### Propositional Calculus for Boolean Valued Functions. Part II

by
Shunichi Kobayashi, and
Yatsuka Nakamura

Copyright (c) 1999 Association of Mizar Users

MML identifier: BVFUNC_6
[ MML identifier index ]

```environ

vocabulary FUNCT_2, MARGREL1, BVFUNC_1, ZF_LANG, BINARITH;
notation XBOOLE_0, SUBSET_1, FRAENKEL, MARGREL1, VALUAT_1, BINARITH, BVFUNC_1;
constructors BINARITH, BVFUNC_1;
clusters MARGREL1, VALUAT_1, BINARITH, FRAENKEL;
theorems MARGREL1, BINARITH, BVFUNC_1, VALUAT_1;

begin

::Chap. 1  Propositional Calculus

reserve Y for non empty set;

theorem for a,b being Element of Funcs(Y,BOOLEAN) holds
a 'imp' (b 'imp' (a '&' b))=I_el(Y)
proof
let a,b be Element of Funcs(Y,BOOLEAN);
for x being Element of Y holds
Pj(a 'imp' (b 'imp' (a '&' b)),x)=TRUE
proof
let x be Element of Y;
Pj(a 'imp' (b 'imp' (a '&' b)),x)
='not' Pj(a,x) 'or' Pj(b 'imp' (a '&' b),x) by BVFUNC_1:def 11
.='not' Pj(a,x) 'or' ('not' Pj(b,x) 'or' Pj(a '&' b,x)) by BVFUNC_1:def 11
.='not' Pj(a,x) 'or' ('not'
Pj(b,x) 'or' (Pj(a,x) '&' Pj(b,x))) by VALUAT_1:def 6
.='not' Pj(a,x) 'or' (('not' Pj(b,x) 'or' Pj(a,x)) '&' ('not'
Pj(b,x) 'or' Pj(b,x)))
by BINARITH:23
.='not' Pj(a,x) 'or' (TRUE '&' ('not' Pj(b,x) 'or' Pj(a,x))) by BINARITH:18
.='not' Pj(a,x) 'or' (Pj(a,x) 'or' 'not' Pj(b,x)) by MARGREL1:50
.=('not' Pj(a,x) 'or' Pj(a,x)) 'or' 'not' Pj(b,x)
by BINARITH:20
.=TRUE 'or' 'not' Pj(b,x)
by BINARITH:18
.=TRUE by BINARITH:19;
hence thesis;
end;
hence thesis by BVFUNC_1:def 14;
end;

theorem for a,b being Element of Funcs(Y,BOOLEAN) holds
(a 'imp' b) 'imp' ((b 'imp' a) 'imp' (a 'eqv' b))=I_el(Y)
proof
let a,b be Element of Funcs(Y,BOOLEAN);
for x being Element of Y holds
Pj((a 'imp' b) 'imp' ((b 'imp' a) 'imp' (a 'eqv' b)),x)=TRUE
proof
let x be Element of Y;
Pj((a 'imp' b) 'imp' ((b 'imp' a) 'imp' (a 'eqv' b)),x)
='not' Pj(a 'imp' b,x) 'or' Pj((b 'imp' a) 'imp' (a 'eqv' b),x)
by BVFUNC_1:def 11
.='not'( 'not' Pj(a,x) 'or' Pj(b,x)) 'or'
Pj((b 'imp' a) 'imp' (a 'eqv' b),x)
by BVFUNC_1:def 11
.='not'( 'not' Pj(a,x) 'or' Pj(b,x)) 'or' ('not'
Pj(b 'imp' a,x) 'or' Pj(a 'eqv' b,x))
by BVFUNC_1:def 11
.='not'( 'not' Pj(a,x) 'or' Pj(b,x)) 'or'
('not'( 'not' Pj(b,x) 'or' Pj(a,x)) 'or' Pj(a 'eqv' b,x))
by BVFUNC_1:def 11
.=('not' 'not' Pj(a,x) '&' 'not' Pj(b,x)) 'or'
('not'( 'not' Pj(b,x) 'or' Pj(a,x)) 'or' Pj(a 'eqv' b,x))
by BINARITH:10
.=('not' 'not' Pj(a,x) '&' 'not' Pj(b,x)) 'or'
(('not' 'not' Pj(b,x) '&' 'not' Pj(a,x)) 'or' Pj(a 'eqv' b,x))
by BINARITH:10
.=(Pj(a,x) '&' 'not' Pj(b,x)) 'or'
(('not' 'not' Pj(b,x) '&' 'not' Pj(a,x)) 'or' Pj(a 'eqv' b,x))
by MARGREL1:40
.=(Pj(a,x) '&' 'not' Pj(b,x)) 'or'
((Pj(b,x) '&' 'not' Pj(a,x)) 'or' Pj(a 'eqv' b,x))
by MARGREL1:40
.=(Pj(a,x) '&' 'not' Pj(b,x)) 'or' ((Pj(b,x) '&' 'not' Pj(a,x)) 'or'
'not'( Pj(a,x) 'xor' Pj(b,x)))
by BVFUNC_1:def 12
.=(Pj(a,x) '&' 'not' Pj(b,x)) 'or' ((Pj(b,x) '&' 'not' Pj(a,x)) 'or'
'not'( ('not' Pj(a,x) '&' Pj(b,x)) 'or' (Pj(a,x) '&' 'not' Pj(b,x))))
by BINARITH:def 2
.=(Pj(a,x) '&' 'not' Pj(b,x)) 'or' (('not' Pj(a,x) '&' Pj(b,x)) 'or'
('not'( 'not' Pj(a,x) '&' Pj(b,x)) '&' 'not'( Pj(a,x) '&' 'not'
Pj(b,x)))) by BINARITH:10
.=(Pj(a,x) '&' 'not' Pj(b,x)) 'or'
((('not' Pj(a,x) '&' Pj(b,x)) 'or' 'not'( 'not' Pj(a,x) '&' Pj(b,x))) '&'
(('not' Pj(a,x) '&' Pj(b,x)) 'or' 'not'( Pj(a,x) '&' 'not' Pj(b,x))))
by BINARITH:23
.=(Pj(a,x) '&' 'not' Pj(b,x)) 'or'
(TRUE '&'
(('not' Pj(a,x) '&' Pj(b,x)) 'or' 'not'( Pj(a,x) '&' 'not' Pj(b,x))))
by BINARITH:18
.=(Pj(a,x) '&' 'not' Pj(b,x)) 'or'
('not'( Pj(a,x) '&' 'not' Pj(b,x)) 'or' ('not'
Pj(a,x) '&' Pj(b,x))) by MARGREL1:50
.=((Pj(a,x) '&' 'not' Pj(b,x)) 'or'
'not'( Pj(a,x) '&' 'not' Pj(b,x))) 'or' ('not' Pj(a,x) '&' Pj(b,x))
by BINARITH:20
.=TRUE 'or' ('not' Pj(a,x) '&' Pj(b,x))
by BINARITH:18
.=TRUE by BINARITH:19;
hence thesis;
end;
hence thesis by BVFUNC_1:def 14;
end;

theorem for a,b being Element of Funcs(Y,BOOLEAN) holds
(a 'or' b) 'eqv' (b 'or' a)=I_el(Y)
proof
let a,b be Element of Funcs(Y,BOOLEAN);
for x being Element of Y holds
Pj((a 'or' b) 'eqv' (b 'or' a),x)=TRUE
proof
let x be Element of Y;
Pj((a 'or' b) 'eqv' (b 'or' a),x)
='not'( Pj(a 'or' b,x) 'xor' Pj(b 'or' a,x))
by BVFUNC_1:def 12
.='not'( ('not' Pj(a 'or' b,x) '&' Pj(b 'or' a,x)) 'or'
(Pj(a 'or' b,x) '&' 'not' Pj(b 'or' a,x)))
by BINARITH:def 2
.=('not'( 'not' Pj(a 'or' b,x) '&' Pj(b 'or' a,x)) '&'
'not'( Pj(a 'or' b,x) '&' 'not' Pj(b 'or' a,x)))
by BINARITH:10
.=(('not' 'not' Pj(a 'or' b,x) 'or' 'not' Pj(b 'or' a,x)) '&'
'not'( Pj(a 'or' b,x) '&' 'not' Pj(b 'or' a,x)))
by BINARITH:9
.=(('not' 'not' Pj(a 'or' b,x) 'or' 'not' Pj(b 'or' a,x)) '&'
('not' Pj(a 'or' b,x) 'or' 'not' 'not' Pj(b 'or' a,x)))
by BINARITH:9
.=((Pj(a 'or' b,x) 'or' 'not' Pj(b 'or' a,x)) '&'
('not' Pj(a 'or' b,x) 'or' 'not' 'not' Pj(b 'or' a,x)))
by MARGREL1:40
.=TRUE '&'
('not' Pj(a 'or' b,x) 'or' Pj(a 'or' b,x))
by BINARITH:18
.=TRUE '&' TRUE by BINARITH:18
.=TRUE by MARGREL1:45;
hence thesis;
end;
hence thesis by BVFUNC_1:def 14;
end;

theorem for a,b,c being Element of Funcs(Y,BOOLEAN) holds
((a '&' b) 'imp' c) 'imp' (a 'imp' (b 'imp' c))=I_el(Y)
proof
let a,b,c be Element of Funcs(Y,BOOLEAN);
for x being Element of Y holds
Pj(((a '&' b) 'imp' c) 'imp' (a 'imp' (b 'imp' c)),x)=TRUE
proof
let x be Element of Y;
Pj(((a '&' b) 'imp' c) 'imp' (a 'imp' (b 'imp' c)),x)
='not' Pj((a '&' b) 'imp' c,x) 'or' Pj(a 'imp' (b 'imp' c),x)
by BVFUNC_1:def 11
.='not'( 'not' Pj(a '&' b,x) 'or' Pj(c,x)) 'or' Pj(a 'imp' (b 'imp' c),x)
by BVFUNC_1:def 11
.='not'( 'not'
(Pj(a,x) '&' Pj(b,x)) 'or' Pj(c,x)) 'or' Pj(a 'imp' (b 'imp' c),x)
by VALUAT_1:def 6
.='not'( 'not'( Pj(a,x) '&' Pj(b,x)) 'or' Pj(c,x)) 'or'
('not' Pj(a,x) 'or' Pj(b 'imp' c,x))
by BVFUNC_1:def 11
.='not'( 'not'( Pj(a,x) '&' Pj(b,x)) 'or' Pj(c,x)) 'or'
('not' Pj(a,x) 'or' ('not' Pj(b,x) 'or' Pj(c,x)))
by BVFUNC_1:def 11
.='not'( ('not' Pj(a,x) 'or' 'not' Pj(b,x)) 'or' Pj(c,x)) 'or'
('not' Pj(a,x) 'or' ('not' Pj(b,x) 'or' Pj(c,x)))
by BINARITH:9
.='not'( ('not' Pj(a,x) 'or' 'not' Pj(b,x)) 'or' Pj(c,x)) 'or'
(('not' Pj(a,x) 'or' 'not' Pj(b,x)) 'or' Pj(c,x))
by BINARITH:20
.=TRUE by BINARITH:18;
hence thesis;
end;
hence thesis by BVFUNC_1:def 14;
end;

theorem for a,b,c being Element of Funcs(Y,BOOLEAN) holds
(a 'imp' (b 'imp' c)) 'imp' ((a '&' b) 'imp' c)=I_el(Y)
proof
let a,b,c be Element of Funcs(Y,BOOLEAN);
for x being Element of Y holds
Pj((a 'imp' (b 'imp' c)) 'imp' ((a '&' b) 'imp' c),x)=TRUE
proof
let x be Element of Y;
Pj((a 'imp' (b 'imp' c)) 'imp' ((a '&' b) 'imp' c),x)
='not' Pj(a 'imp' (b 'imp' c),x) 'or' Pj((a '&' b) 'imp' c,x)
by BVFUNC_1:def 11
.='not'( 'not' Pj(a,x) 'or' Pj(b 'imp' c,x)) 'or'
Pj((a '&' b) 'imp' c,x)
by BVFUNC_1:def 11
.='not'( 'not' Pj(a,x) 'or' ('not'
Pj(b,x) 'or' Pj(c,x))) 'or' Pj((a '&' b) 'imp' c,x)
by BVFUNC_1:def 11
.='not'( 'not' Pj(a,x) 'or' ('not' Pj(b,x) 'or' Pj(c,x))) 'or'
('not' Pj(a '&' b,x) 'or' Pj(c,x))
by BVFUNC_1:def 11
.='not'( 'not' Pj(a,x) 'or' ('not' Pj(b,x) 'or' Pj(c,x))) 'or'
('not'( Pj(a,x) '&' Pj(b,x)) 'or' Pj(c,x))
by VALUAT_1:def 6
.='not'( 'not' Pj(a,x) 'or' ('not' Pj(b,x) 'or' Pj(c,x))) 'or'
(('not' Pj(a,x) 'or' 'not' Pj(b,x)) 'or' Pj(c,x))
by BINARITH:9
.='not'( ('not' Pj(a,x) 'or' 'not' Pj(b,x)) 'or' Pj(c,x)) 'or'
(('not' Pj(a,x) 'or' 'not' Pj(b,x)) 'or' Pj(c,x))
by BINARITH:20
.=TRUE by BINARITH:18;
hence thesis;
end;
hence thesis by BVFUNC_1:def 14;
end;

theorem for a,b,c being Element of Funcs(Y,BOOLEAN) holds
(c 'imp' a) 'imp' ((c 'imp' b) 'imp' (c 'imp' (a '&' b)))=I_el(Y)
proof
let a,b,c be Element of Funcs(Y,BOOLEAN);
for x being Element of Y holds
Pj((c 'imp' a) 'imp' ((c 'imp' b) 'imp' (c 'imp' (a '&' b))),x)=TRUE
proof
let x be Element of Y;
Pj((c 'imp' a) 'imp' ((c 'imp' b) 'imp' (c 'imp' (a '&' b))),x)
='not' Pj(c 'imp' a,x) 'or' Pj((c 'imp' b) 'imp' (c 'imp' (a '&' b)),x)
by BVFUNC_1:def 11
.='not'( 'not' Pj(c,x) 'or' Pj(a,x)) 'or'
Pj((c 'imp' b) 'imp' (c 'imp' (a '&' b)),x)
by BVFUNC_1:def 11
.='not'( 'not' Pj(c,x) 'or' Pj(a,x)) 'or'
('not' Pj(c 'imp' b,x) 'or' Pj(c 'imp' (a '&' b),x))
by BVFUNC_1:def 11
.='not'( 'not' Pj(c,x) 'or' Pj(a,x)) 'or'
('not'( 'not' Pj(c,x) 'or' Pj(b,x)) 'or' Pj(c 'imp' (a '&' b),x))
by BVFUNC_1:def 11
.='not'( 'not' Pj(c,x) 'or' Pj(a,x)) 'or'
('not'( 'not' Pj(c,x) 'or' Pj(b,x)) 'or' ('not' Pj(c,x) 'or'
Pj(a '&' b,x)))
by BVFUNC_1:def 11
.='not'( 'not' Pj(c,x) 'or' Pj(a,x)) 'or'
('not'( 'not' Pj(c,x) 'or' Pj(b,x)) 'or' ('not'
Pj(c,x) 'or' (Pj(a,x) '&' Pj(b,x))))
by VALUAT_1:def 6
.=('not' 'not' Pj(c,x) '&' 'not' Pj(a,x)) 'or'
('not'( 'not' Pj(c,x) 'or' Pj(b,x)) 'or' ('not'
Pj(c,x) 'or' (Pj(a,x) '&' Pj(b,x))))
by BINARITH:10
.=('not' 'not' Pj(c,x) '&' 'not' Pj(a,x)) 'or'
(('not' 'not' Pj(c,x) '&' 'not' Pj(b,x)) 'or' ('not'
Pj(c,x) 'or' (Pj(a,x) '&' Pj(b,x))))
by BINARITH:10
.=(Pj(c,x) '&' 'not' Pj(a,x)) 'or'
(('not' 'not' Pj(c,x) '&' 'not' Pj(b,x)) 'or' ('not'
Pj(c,x) 'or' (Pj(a,x) '&' Pj(b,x))))
by MARGREL1:40
.=(Pj(c,x) '&' 'not' Pj(a,x)) 'or'
((Pj(c,x) '&' 'not' Pj(b,x)) 'or' ('not'
Pj(c,x) 'or' (Pj(a,x) '&' Pj(b,x))))
by MARGREL1:40
.=(Pj(c,x) '&' 'not' Pj(a,x)) 'or'
((Pj(c,x) '&' 'not' Pj(b,x)) 'or'
(('not' Pj(c,x) 'or' Pj(a,x)) '&' ('not' Pj(c,x) 'or' Pj(b,x))))
by BINARITH:23
.=(Pj(c,x) '&' 'not' Pj(a,x)) 'or'
(((Pj(c,x) '&' 'not' Pj(b,x)) 'or' ('not' Pj(c,x) 'or' Pj(a,x))) '&'
((Pj(c,x) '&' 'not' Pj(b,x)) 'or' ('not' Pj(c,x) 'or' Pj(b,x))))
by BINARITH:23
.=(Pj(c,x) '&' 'not' Pj(a,x)) 'or'
(((Pj(c,x) '&' 'not' Pj(b,x)) 'or' ('not' Pj(c,x) 'or' Pj(a,x))) '&'
((Pj(c,x) '&' 'not' Pj(b,x)) 'or' ('not'
Pj(c,x) 'or' 'not' 'not' Pj(b,x))))
by MARGREL1:40
.=(Pj(c,x) '&' 'not' Pj(a,x)) 'or'
(((Pj(c,x) '&' 'not' Pj(b,x)) 'or' ('not' Pj(c,x) 'or' Pj(a,x))) '&'
((Pj(c,x) '&' 'not' Pj(b,x)) 'or' 'not'( Pj(c,x) '&' 'not' Pj(b,x))))
by BINARITH:9
.=(Pj(c,x) '&' 'not' Pj(a,x)) 'or'
(TRUE '&'
((Pj(c,x) '&' 'not' Pj(b,x)) 'or' ('not'
Pj(c,x) 'or' Pj(a,x)))) by BINARITH:18
.=(Pj(c,x) '&' 'not' Pj(a,x)) 'or'
(('not' Pj(c,x) 'or' Pj(a,x)) 'or' (Pj(c,x) '&' 'not'
Pj(b,x))) by MARGREL1:50
.=((Pj(c,x) '&' 'not' Pj(a,x)) 'or' ('not' Pj(c,x) 'or' Pj(a,x))) 'or'
(Pj(c,x) '&' 'not' Pj(b,x))
by BINARITH:20
.=((Pj(c,x) '&' 'not' Pj(a,x)) 'or' ('not' Pj(c,x) 'or' 'not' 'not'
Pj(a,x))) 'or'
(Pj(c,x) '&' 'not' Pj(b,x))
by MARGREL1:40
.=((Pj(c,x) '&' 'not' Pj(a,x)) 'or' 'not'( Pj(c,x) '&' 'not' Pj(a,x))) 'or'
(Pj(c,x) '&' 'not' Pj(b,x))
by BINARITH:9
.=TRUE 'or'
(Pj(c,x) '&' 'not' Pj(b,x))
by BINARITH:18
.=TRUE by BINARITH:19;
hence thesis;
end;
hence thesis by BVFUNC_1:def 14;
end;

theorem for a,b,c being Element of Funcs(Y,BOOLEAN) holds
((a 'or' b) 'imp' c) 'imp' ((a 'imp' c) 'or' (b 'imp' c))=I_el(Y)
proof
let a,b,c be Element of Funcs(Y,BOOLEAN);
for x being Element of Y holds
Pj(((a 'or' b) 'imp' c) 'imp' ((a 'imp' c) 'or' (b 'imp' c)),x)=TRUE
proof
let x be Element of Y;
Pj(((a 'or' b) 'imp' c) 'imp' ((a 'imp' c) 'or' (b 'imp' c)),x)
='not' Pj((a 'or' b) 'imp' c,x) 'or' Pj((a 'imp' c) 'or' (b 'imp' c),x)
by BVFUNC_1:def 11
.='not'( 'not' Pj(a 'or' b,x) 'or' Pj(c,x)) 'or'
Pj((a 'imp' c) 'or' (b 'imp' c),x)
by BVFUNC_1:def 11
.='not'( 'not'( Pj(a,x) 'or' Pj(b,x)) 'or' Pj(c,x)) 'or'
Pj((a 'imp' c) 'or' (b 'imp' c),x)
by BVFUNC_1:def 7
.='not'( 'not'( Pj(a,x) 'or' Pj(b,x)) 'or' Pj(c,x)) 'or'
(Pj(a 'imp' c,x) 'or' Pj(b 'imp' c,x))
by BVFUNC_1:def 7
.='not'( 'not'( Pj(a,x) 'or' Pj(b,x)) 'or' Pj(c,x)) 'or'
(('not' Pj(a,x) 'or' Pj(c,x)) 'or' Pj(b 'imp' c,x))
by BVFUNC_1:def 11
.='not'( 'not'( Pj(a,x) 'or' Pj(b,x)) 'or' Pj(c,x)) 'or'
(('not' Pj(a,x) 'or' Pj(c,x)) 'or' ('not' Pj(b,x) 'or' Pj(c,x)))
by BVFUNC_1:def 11
.=('not' 'not'( Pj(a,x) 'or' Pj(b,x)) '&' 'not' Pj(c,x)) 'or'
(('not' Pj(a,x) 'or' Pj(c,x)) 'or' ('not' Pj(b,x) 'or' Pj(c,x)))
by BINARITH:10
.=('not' Pj(c,x) '&' (Pj(a,x) 'or' Pj(b,x))) 'or'
(('not' Pj(a,x) 'or' Pj(c,x)) 'or' ('not'
Pj(b,x) 'or' Pj(c,x))) by MARGREL1:40
.=((Pj(b,x) '&' 'not' Pj(c,x)) 'or' (Pj(a,x) '&' 'not' Pj(c,x))) 'or'
(('not' Pj(a,x) 'or' Pj(c,x)) 'or' ('not'
Pj(b,x) 'or' Pj(c,x))) by BINARITH:22
.=((Pj(b,x) '&' 'not' Pj(c,x)) 'or' (Pj(a,x) '&' 'not' Pj(c,x))) 'or'
(('not' Pj(a,x) 'or' 'not' 'not' Pj(c,x)) 'or' ('not'
Pj(b,x) 'or' Pj(c,x)))
by MARGREL1:40
.=((Pj(b,x) '&' 'not' Pj(c,x)) 'or' (Pj(a,x) '&' 'not' Pj(c,x))) 'or'
('not'( Pj(a,x) '&' 'not' Pj(c,x)) 'or' ('not' Pj(b,x) 'or' Pj(c,x)))
by BINARITH:9
.=(((Pj(b,x) '&' 'not' Pj(c,x)) 'or' (Pj(a,x) '&' 'not' Pj(c,x))) 'or'
'not'( Pj(a,x) '&' 'not' Pj(c,x))) 'or' ('not' Pj(b,x) 'or' Pj(c,x))
by BINARITH:20
.=((Pj(b,x) '&' 'not' Pj(c,x)) 'or' ((Pj(a,x) '&' 'not' Pj(c,x)) 'or'
'not'( Pj(a,x) '&' 'not' Pj(c,x)))) 'or' ('not' Pj(b,x) 'or' Pj(c,x))
by BINARITH:20
.=((Pj(b,x) '&' 'not' Pj(c,x)) 'or' TRUE) 'or' ('not' Pj(b,x) 'or' Pj(c,x))
by BINARITH:18
.=TRUE 'or' ('not' Pj(b,x) 'or' Pj(c,x))
by BINARITH:19
.=TRUE by BINARITH:19;
hence thesis;
end;
hence thesis by BVFUNC_1:def 14;
end;

theorem for a,b,c being Element of Funcs(Y,BOOLEAN) holds
(a 'imp' c) 'imp' ((b 'imp' c) 'imp' ((a 'or' b) 'imp' c))=I_el(Y)
proof
let a,b,c be Element of Funcs(Y,BOOLEAN);
for x being Element of Y holds
Pj((a 'imp' c) 'imp' ((b 'imp' c) 'imp' ((a 'or' b) 'imp' c)),x)=TRUE
proof
let x be Element of Y;
Pj((a 'imp' c) 'imp' ((b 'imp' c) 'imp' ((a 'or' b) 'imp' c)),x)
='not' Pj(a 'imp' c,x) 'or' Pj((b 'imp' c) 'imp' ((a 'or' b) 'imp' c),x)
by BVFUNC_1:def 11
.='not'( 'not' Pj(a,x) 'or' Pj(c,x)) 'or'
Pj((b 'imp' c) 'imp' ((a 'or' b) 'imp' c),x)
by BVFUNC_1:def 11
.='not'( 'not' Pj(a,x) 'or' Pj(c,x)) 'or'
('not' Pj(b 'imp' c,x) 'or' Pj((a 'or' b) 'imp' c,x))
by BVFUNC_1:def 11
.='not'( 'not' Pj(a,x) 'or' Pj(c,x)) 'or'
('not'( 'not' Pj(b,x) 'or' Pj(c,x)) 'or' Pj((a 'or' b) 'imp' c,x))
by BVFUNC_1:def 11
.='not'( 'not' Pj(a,x) 'or' Pj(c,x)) 'or'
('not'( 'not' Pj(b,x) 'or' Pj(c,x)) 'or' ('not' Pj(a 'or' b,x) 'or'
Pj(c,x)))
by BVFUNC_1:def 11
.='not'( 'not' Pj(a,x) 'or' Pj(c,x)) 'or'
('not'( 'not' Pj(b,x) 'or' Pj(c,x)) 'or' ('not'
(Pj(a,x) 'or' Pj(b,x)) 'or' Pj(c,x)))
by BVFUNC_1:def 7
.='not'( 'not' Pj(a,x) 'or' Pj(c,x)) 'or'
('not'( 'not' Pj(b,x) 'or' Pj(c,x)) 'or' (Pj(c,x) 'or' ('not' Pj(a,x) '&'
'not' Pj(b,x)))) by BINARITH:10
.='not'( 'not' Pj(a,x) 'or' Pj(c,x)) 'or'
('not'( 'not' Pj(b,x) 'or' Pj(c,x)) 'or'
((Pj(c,x) 'or' 'not' Pj(a,x)) '&' ('not'
Pj(b,x) 'or' Pj(c,x)))) by BINARITH:23
.='not'( 'not' Pj(a,x) 'or' Pj(c,x)) 'or'
(('not'( 'not' Pj(b,x) 'or' Pj(c,x)) 'or' (Pj(c,x) 'or' 'not'
Pj(a,x))) '&'
('not'( 'not' Pj(b,x) 'or' Pj(c,x)) 'or' ('not' Pj(b,x) 'or'
Pj(c,x))))
by BINARITH:23
.='not'( 'not' Pj(a,x) 'or' Pj(c,x)) 'or'
(TRUE '&'
('not'( 'not' Pj(b,x) 'or' Pj(c,x)) 'or' (Pj(c,x) 'or' 'not'
Pj(a,x)))) by BINARITH:18
.='not'( 'not' Pj(a,x) 'or' Pj(c,x)) 'or'
('not'( 'not' Pj(b,x) 'or' Pj(c,x)) 'or' ('not'
Pj(a,x) 'or' Pj(c,x))) by MARGREL1:50
.=('not'( 'not' Pj(a,x) 'or' Pj(c,x)) 'or'
('not' Pj(a,x) 'or' Pj(c,x))) 'or' 'not'( 'not' Pj(b,x) 'or' Pj(c,x))
by BINARITH:20
.=TRUE 'or' 'not'( 'not' Pj(b,x) 'or' Pj(c,x))
by BINARITH:18
.=TRUE by BINARITH:19;
hence thesis;
end;
hence thesis by BVFUNC_1:def 14;
end;

theorem for a,b,c being Element of Funcs(Y,BOOLEAN) holds
((a 'imp' c) '&' (b 'imp' c)) 'imp' ((a 'or' b) 'imp' c)=I_el(Y)
proof
let a,b,c be Element of Funcs(Y,BOOLEAN);
for x being Element of Y holds
Pj(((a 'imp' c) '&' (b 'imp' c)) 'imp' ((a 'or' b) 'imp' c),x)=TRUE
proof
let x be Element of Y;
Pj(((a 'imp' c) '&' (b 'imp' c)) 'imp' ((a 'or' b) 'imp' c),x)
='not' Pj((a 'imp' c) '&' (b 'imp' c),x) 'or' Pj((a 'or' b) 'imp' c,x)
by BVFUNC_1:def 11
.='not'( Pj(a 'imp' c,x) '&' Pj(b 'imp' c,x)) 'or' Pj((a 'or' b) 'imp' c,x)
by VALUAT_1:def 6
.='not'( ('not' Pj(a,x) 'or' Pj(c,x)) '&' Pj(b 'imp' c,x)) 'or'
Pj((a 'or' b) 'imp' c,x) by BVFUNC_1:def 11
.='not'( ('not' Pj(a,x) 'or' Pj(c,x)) '&' ('not' Pj(b,x) 'or' Pj(c,x))) 'or'
Pj((a 'or' b) 'imp' c,x) by BVFUNC_1:def 11
.='not'( ('not' Pj(a,x) 'or' Pj(c,x)) '&' ('not' Pj(b,x) 'or' Pj(c,x))) 'or'
('not' Pj(a 'or' b,x) 'or' Pj(c,x))
by BVFUNC_1:def 11
.='not'( ('not' Pj(a,x) 'or' Pj(c,x)) '&' ('not' Pj(b,x) 'or' Pj(c,x))) 'or'
('not'( Pj(a,x) 'or' Pj(b,x)) 'or' Pj(c,x))
by BVFUNC_1:def 7
.=('not'( 'not' Pj(a,x) 'or' Pj(c,x)) 'or' 'not'( 'not'
Pj(b,x) 'or' Pj(c,x))) 'or'
(Pj(c,x) 'or' 'not'( Pj(a,x) 'or' Pj(b,x))) by BINARITH:9
.=('not'( 'not' Pj(a,x) 'or' Pj(c,x)) 'or' 'not'( 'not'
Pj(b,x) 'or' Pj(c,x))) 'or'
(Pj(c,x) 'or' ('not' Pj(a,x) '&' 'not' Pj(b,x))) by BINARITH:10
.=('not'( 'not' Pj(a,x) 'or' Pj(c,x)) 'or' 'not'( 'not'
Pj(b,x) 'or' Pj(c,x))) 'or'
(('not' Pj(a,x) 'or' Pj(c,x)) '&' (Pj(c,x) 'or' 'not'
Pj(b,x))) by BINARITH:23
.=(('not'( 'not' Pj(a,x) 'or' Pj(c,x)) 'or' 'not'( 'not'
Pj(b,x) 'or' Pj(c,x))) 'or'
('not' Pj(a,x) 'or' Pj(c,x))) '&'
(('not'( 'not' Pj(a,x) 'or' Pj(c,x)) 'or' 'not'( 'not'
Pj(b,x) 'or' Pj(c,x))) 'or'
('not' Pj(b,x) 'or' Pj(c,x))) by BINARITH:23
.=('not'( 'not' Pj(b,x) 'or' Pj(c,x)) 'or' ('not'( 'not'
Pj(a,x) 'or' Pj(c,x)) 'or'
('not' Pj(a,x) 'or' Pj(c,x)))) '&'
(('not'( 'not' Pj(a,x) 'or' Pj(c,x)) 'or' 'not'( 'not'
Pj(b,x) 'or' Pj(c,x))) 'or'
('not' Pj(b,x) 'or' Pj(c,x)))
by BINARITH:20
.=('not'( 'not' Pj(b,x) 'or' Pj(c,x)) 'or' ('not'( 'not'
Pj(a,x) 'or' Pj(c,x)) 'or'
('not' Pj(a,x) 'or' Pj(c,x)))) '&'
('not'( 'not' Pj(a,x) 'or' Pj(c,x)) 'or' ('not'( 'not'
Pj(b,x) 'or' Pj(c,x)) 'or'
('not' Pj(b,x) 'or' Pj(c,x))))
by BINARITH:20
.=('not'( 'not' Pj(b,x) 'or' Pj(c,x)) 'or' TRUE) '&'
('not'( 'not' Pj(a,x) 'or' Pj(c,x)) 'or' ('not'( 'not'
Pj(b,x) 'or' Pj(c,x)) 'or'
('not' Pj(b,x) 'or' Pj(c,x))))
by BINARITH:18
.=('not'( 'not' Pj(b,x) 'or' Pj(c,x)) 'or' TRUE) '&'
('not'( 'not' Pj(a,x) 'or' Pj(c,x)) 'or' TRUE) by BINARITH:18
.=TRUE '&' ('not'( 'not' Pj(a,x) 'or' Pj(c,x)) 'or' TRUE) by BINARITH:19
.=TRUE '&' TRUE by BINARITH:19
.=TRUE by MARGREL1:45;
hence thesis;
end;
hence thesis by BVFUNC_1:def 14;
end;

theorem for a,b being Element of Funcs(Y,BOOLEAN) holds
(a 'imp' (b '&' 'not' b)) 'imp' 'not' a=I_el(Y)
proof
let a,b be Element of Funcs(Y,BOOLEAN);
for x being Element of Y holds
Pj((a 'imp' (b '&' 'not' b)) 'imp' 'not' a,x)=TRUE
proof
let x be Element of Y;
Pj((a 'imp' (b '&' 'not' b)) 'imp' 'not' a,x)
='not' Pj(a 'imp' (b '&' 'not' b),x) 'or' Pj('not' a,x) by BVFUNC_1:def 11
.='not'( 'not' Pj(a,x) 'or' Pj(b '&' 'not' b,x)) 'or' Pj('not'
a,x) by BVFUNC_1:def 11
.='not'( 'not' Pj(a,x) 'or' (Pj(b,x) '&' Pj('not' b,x))) 'or' Pj('not'
a,x) by VALUAT_1:def 6
.=('not' 'not' Pj(a,x) '&' 'not'( Pj(b,x) '&' Pj('not' b,x))) 'or' Pj('not'
a,x) by BINARITH:10
.=(Pj(a,x) '&' 'not'( Pj(b,x) '&' Pj('not' b,x))) 'or' Pj('not'
a,x) by MARGREL1:40
.=(Pj(a,x) '&' ('not' Pj(b,x) 'or' 'not' Pj('not' b,x))) 'or' Pj('not'
a,x) by BINARITH:9
.=(Pj(a,x) '&' ('not' Pj(b,x) 'or' 'not' 'not' Pj(b,x))) 'or' Pj('not'
a,x) by VALUAT_1:def 5
.=(Pj(a,x) '&' TRUE) 'or' Pj('not' a,x) by BINARITH:18
.=Pj(a,x) 'or' Pj('not' a,x) by MARGREL1:50
.=Pj(a,x) 'or' 'not' Pj(a,x) by VALUAT_1:def 5
.=TRUE by BINARITH:18;
hence thesis;
end;
hence thesis by BVFUNC_1:def 14;
end;

theorem for a,b,c being Element of Funcs(Y,BOOLEAN) holds
((a 'or' b) '&' (a 'or' c)) 'imp' (a 'or' (b '&' c))=I_el(Y)
proof
let a,b,c be Element of Funcs(Y,BOOLEAN);
for x being Element of Y holds
Pj(((a 'or' b) '&' (a 'or' c)) 'imp' (a 'or' (b '&' c)),x)=TRUE
proof
let x be Element of Y;
Pj(((a 'or' b) '&' (a 'or' c)) 'imp' (a 'or' (b '&' c)),x)
='not' Pj((a 'or' b) '&' (a 'or' c),x) 'or' Pj(a 'or' (b '&' c),x)
by BVFUNC_1:def 11
.='not'( Pj(a 'or' b,x) '&' Pj(a 'or' c,x)) 'or' Pj(a 'or' (b '&' c),x)
by VALUAT_1:def 6
.='not'
((Pj(a,x) 'or' Pj(b,x)) '&' Pj(a 'or' c,x)) 'or' Pj(a 'or' (b '&' c),x)
by BVFUNC_1:def 7
.='not'( (Pj(a,x) 'or' Pj(b,x)) '&' (Pj(a,x) 'or' Pj(c,x))) 'or'
Pj(a 'or' (b '&' c),x)
by BVFUNC_1:def 7
.='not'( (Pj(a,x) 'or' Pj(b,x)) '&' (Pj(a,x) 'or' Pj(c,x))) 'or'
(Pj(a,x) 'or' Pj(b '&' c,x))
by BVFUNC_1:def 7
.='not'( (Pj(a,x) 'or' Pj(b,x)) '&' (Pj(a,x) 'or' Pj(c,x))) 'or'
(Pj(a,x) 'or' (Pj(b,x) '&' Pj(c,x)))
by VALUAT_1:def 6
.='not'( (Pj(a,x) 'or' Pj(b,x)) '&' (Pj(a,x) 'or' Pj(c,x))) 'or'
((Pj(a,x) 'or' Pj(b,x)) '&' (Pj(a,x) 'or' Pj(c,x)))
by BINARITH:23
.=('not'( Pj(a,x) 'or' Pj(b,x)) 'or' 'not'( Pj(a,x) 'or' Pj(c,x))) 'or'
((Pj(a,x) 'or' Pj(b,x)) '&' (Pj(a,x) 'or' Pj(c,x)))
by BINARITH:9
.=(('not'( Pj(a,x) 'or' Pj(c,x)) 'or' 'not'( Pj(a,x) 'or' Pj(b,x))) 'or'
(Pj(a,x) 'or' Pj(b,x))) '&'
(('not'( Pj(a,x) 'or' Pj(b,x)) 'or' 'not'( Pj(a,x) 'or' Pj(c,x))) 'or'
(Pj(a,x) 'or' Pj(c,x))) by BINARITH:23
.=('not'( Pj(a,x) 'or' Pj(c,x)) 'or' ('not'( Pj(a,x) 'or' Pj(b,x)) 'or'
(Pj(a,x) 'or' Pj(b,x)))) '&'
(('not'( Pj(a,x) 'or' Pj(b,x)) 'or' 'not'( Pj(a,x) 'or' Pj(c,x))) 'or'
(Pj(a,x) 'or' Pj(c,x)))
by BINARITH:20
.=('not'( Pj(a,x) 'or' Pj(c,x)) 'or' ('not'( Pj(a,x) 'or' Pj(b,x)) 'or'
(Pj(a,x) 'or' Pj(b,x)))) '&'
('not'( Pj(a,x) 'or' Pj(b,x)) 'or' ('not'( Pj(a,x) 'or' Pj(c,x)) 'or'
(Pj(a,x) 'or' Pj(c,x))))
by BINARITH:20
.=('not'( Pj(a,x) 'or' Pj(c,x)) 'or' TRUE) '&'
('not'( Pj(a,x) 'or' Pj(b,x)) 'or' ('not'( Pj(a,x) 'or' Pj(c,x)) 'or'
(Pj(a,x) 'or' Pj(c,x))))
by BINARITH:18
.=('not'( Pj(a,x) 'or' Pj(c,x)) 'or' TRUE) '&'
('not'( Pj(a,x) 'or' Pj(b,x)) 'or' TRUE) by BINARITH:18
.=TRUE '&' ('not'( Pj(a,x) 'or' Pj(b,x)) 'or' TRUE) by BINARITH:19
.=TRUE '&' TRUE by BINARITH:19
.=TRUE by MARGREL1:45;
hence thesis;
end;
hence thesis by BVFUNC_1:def 14;
end;

theorem for a,b,c being Element of Funcs(Y,BOOLEAN) holds
(a '&' (b 'or' c)) 'imp' ((a '&' b) 'or' (a '&' c))=I_el(Y)
proof
let a,b,c be Element of Funcs(Y,BOOLEAN);
for x being Element of Y holds
Pj((a '&' (b 'or' c)) 'imp' ((a '&' b) 'or' (a '&' c)),x)=TRUE
proof
let x be Element of Y;
Pj((a '&' (b 'or' c)) 'imp' ((a '&' b) 'or' (a '&' c)),x)
='not' Pj(a '&' (b 'or' c),x) 'or' Pj((a '&' b) 'or' (a '&' c),x)
by BVFUNC_1:def 11
.='not'( Pj(a,x) '&' Pj(b 'or' c,x)) 'or' Pj((a '&' b) 'or' (a '&' c),x)
by VALUAT_1:def 6
.='not'
(Pj(a,x) '&' (Pj(b,x) 'or' Pj(c,x))) 'or' Pj((a '&' b) 'or' (a '&' c),x)
by BVFUNC_1:def 7
.='not'( Pj(a,x) '&' (Pj(b,x) 'or' Pj(c,x))) 'or'
(Pj(a '&' b,x) 'or' Pj(a '&' c,x)) by BVFUNC_1:def 7
.='not'( Pj(a,x) '&' (Pj(b,x) 'or' Pj(c,x))) 'or'
((Pj(a,x) '&' Pj(b,x)) 'or' Pj(a '&' c,x))
by VALUAT_1:def 6
.='not'( Pj(a,x) '&' (Pj(b,x) 'or' Pj(c,x))) 'or'
((Pj(a,x) '&' Pj(b,x)) 'or' (Pj(a,x) '&' Pj(c,x)))
by VALUAT_1:def 6
.='not'( (Pj(a,x) '&' Pj(b,x)) 'or' (Pj(a,x) '&' Pj(c,x))) 'or'
((Pj(a,x) '&' Pj(b,x)) 'or' (Pj(a,x) '&' Pj(c,x)))
by BINARITH:22
.= ((Pj(a,x) '&' Pj(b,x)) 'or' (Pj(a,x) '&' Pj(c,x))) 'or'
('not'( Pj(a,x) '&' Pj(b,x)) '&' 'not'
(Pj(a,x) '&' Pj(c,x))) by BINARITH:10
.= (((Pj(a,x) '&' Pj(c,x)) 'or' (Pj(a,x) '&' Pj(b,x))) 'or'
'not'( Pj(a,x) '&' Pj(b,x))) '&'
(((Pj(a,x) '&' Pj(b,x)) 'or' (Pj(a,x) '&' Pj(c,x))) 'or'
'not'( Pj(a,x) '&' Pj(c,x))) by BINARITH:23
.= ((Pj(a,x) '&' Pj(c,x)) 'or' ((Pj(a,x) '&' Pj(b,x)) 'or'
'not'( Pj(a,x) '&' Pj(b,x)))) '&'
(((Pj(a,x) '&' Pj(b,x)) 'or' (Pj(a,x) '&' Pj(c,x))) 'or'
'not'( Pj(a,x) '&' Pj(c,x))) by BINARITH:20
.= ((Pj(a,x) '&' Pj(c,x)) 'or' ((Pj(a,x) '&' Pj(b,x)) 'or'
'not'( Pj(a,x) '&' Pj(b,x)))) '&'
((Pj(a,x) '&' Pj(b,x)) 'or' ((Pj(a,x) '&' Pj(c,x)) 'or'
'not'( Pj(a,x) '&' Pj(c,x)))) by BINARITH:20
.= ((Pj(a,x) '&' Pj(c,x)) 'or' TRUE) '&'
((Pj(a,x) '&' Pj(b,x)) 'or' ((Pj(a,x) '&' Pj(c,x)) 'or'
'not'( Pj(a,x) '&' Pj(c,x))))
by BINARITH:18
.= ((Pj(a,x) '&' Pj(c,x)) 'or' TRUE) '&'
((Pj(a,x) '&' Pj(b,x)) 'or' TRUE)
by BINARITH:18
.= TRUE '&' ((Pj(a,x) '&' Pj(b,x)) 'or' TRUE) by BINARITH:19
.= TRUE '&' TRUE by BINARITH:19
.= TRUE by MARGREL1:45;
hence thesis;
end;
hence thesis by BVFUNC_1:def 14;
end;

theorem for a,b,c being Element of Funcs(Y,BOOLEAN) holds
((a 'or' c) '&' (b 'or' c)) 'imp' ((a '&' b) 'or' c)=I_el(Y)
proof
let a,b,c be Element of Funcs(Y,BOOLEAN);
for x being Element of Y holds
Pj(((a 'or' c) '&' (b 'or' c)) 'imp' ((a '&' b) 'or' c),x)=TRUE
proof
let x be Element of Y;
Pj(((a 'or' c) '&' (b 'or' c)) 'imp' ((a '&' b) 'or' c),x)
='not' Pj((a 'or' c) '&' (b 'or' c),x) 'or' Pj((a '&' b) 'or' c,x)
by BVFUNC_1:def 11
.='not'( Pj(a 'or' c,x) '&' Pj(b 'or' c,x)) 'or' Pj((a '&' b) 'or' c,x)
by VALUAT_1:def 6
.='not'
((Pj(a,x) 'or' Pj(c,x)) '&' Pj(b 'or' c,x)) 'or' Pj((a '&' b) 'or' c,x)
by BVFUNC_1:def 7
.='not'( (Pj(a,x) 'or' Pj(c,x)) '&' (Pj(b,x) 'or' Pj(c,x))) 'or'
Pj((a '&' b) 'or' c,x) by BVFUNC_1:def 7
.=('not'( Pj(a,x) 'or' Pj(c,x)) 'or' 'not'( Pj(b,x) 'or' Pj(c,x))) 'or'
Pj((a '&' b) 'or' c,x) by BINARITH:9
.=('not'( Pj(a,x) 'or' Pj(c,x)) 'or' 'not'( Pj(b,x) 'or' Pj(c,x))) 'or'
(Pj(a '&' b,x) 'or' Pj(c,x)) by BVFUNC_1:def 7
.=('not'( Pj(a,x) 'or' Pj(c,x)) 'or' 'not'( Pj(b,x) 'or' Pj(c,x))) 'or'
(Pj(c,x) 'or' (Pj(a,x) '&' Pj(b,x))) by VALUAT_1:def 6
.=('not'( Pj(a,x) 'or' Pj(c,x)) 'or' 'not'( Pj(b,x) 'or' Pj(c,x))) 'or'
((Pj(a,x) 'or' Pj(c,x)) '&' (Pj(c,x) 'or' Pj(b,x))) by BINARITH:23
.=(('not'( Pj(a,x) 'or' Pj(c,x)) 'or' 'not'( Pj(b,x) 'or' Pj(c,x))) 'or'
(Pj(a,x) 'or' Pj(c,x))) '&'
(('not'( Pj(a,x) 'or' Pj(c,x)) 'or' 'not'( Pj(b,x) 'or' Pj(c,x))) 'or'
(Pj(b,x) 'or' Pj(c,x))) by BINARITH:23
.=('not'( Pj(b,x) 'or' Pj(c,x)) 'or' ('not'( Pj(a,x) 'or' Pj(c,x)) 'or'
(Pj(a,x) 'or' Pj(c,x)))) '&'
(('not'( Pj(a,x) 'or' Pj(c,x)) 'or' 'not'( Pj(b,x) 'or' Pj(c,x))) 'or'
(Pj(b,x) 'or' Pj(c,x)))
by BINARITH:20
.=('not'( Pj(b,x) 'or' Pj(c,x)) 'or' ('not'( Pj(a,x) 'or' Pj(c,x)) 'or'
(Pj(a,x) 'or' Pj(c,x)))) '&'
('not'( Pj(a,x) 'or' Pj(c,x)) 'or' ('not'( Pj(b,x) 'or' Pj(c,x)) 'or'
(Pj(b,x) 'or' Pj(c,x)))) by BINARITH:20
.=('not'( Pj(b,x) 'or' Pj(c,x)) 'or' TRUE) '&'
('not'( Pj(a,x) 'or' Pj(c,x)) 'or' ('not'( Pj(b,x) 'or' Pj(c,x)) 'or'
(Pj(b,x) 'or' Pj(c,x)))) by BINARITH:18
.=('not'( Pj(b,x) 'or' Pj(c,x)) 'or' TRUE) '&'
('not'( Pj(a,x) 'or' Pj(c,x)) 'or' TRUE) by BINARITH:18
.=TRUE '&' ('not'( Pj(a,x) 'or' Pj(c,x)) 'or' TRUE) by BINARITH:19
.=TRUE '&' TRUE by BINARITH:19
.=TRUE by MARGREL1:45;
hence thesis;
end;
hence thesis by BVFUNC_1:def 14;
end;

theorem for a,b,c being Element of Funcs(Y,BOOLEAN) holds
((a 'or' b) '&' c) 'imp' ((a '&' c) 'or' (b '&' c))=I_el(Y)
proof
let a,b,c be Element of Funcs(Y,BOOLEAN);
for x being Element of Y holds
Pj(((a 'or' b) '&' c) 'imp' ((a '&' c) 'or' (b '&' c)),x)=TRUE
proof
let x be Element of Y;
Pj(((a 'or' b) '&' c) 'imp' ((a '&' c) 'or' (b '&' c)),x)
='not' Pj((a 'or' b) '&' c,x) 'or' Pj((a '&' c) 'or' (b '&' c),x)
by BVFUNC_1:def 11
.='not'( Pj((a 'or' b),x) '&' Pj(c,x)) 'or' Pj((a '&' c) 'or' (b '&' c),x)
by VALUAT_1:def 6
.='not'( (Pj(a,x) 'or' Pj(b,x)) '&' Pj(c,x)) 'or'
Pj((a '&' c) 'or' (b '&' c),x) by BVFUNC_1:def 7
.='not'( (Pj(a,x) 'or' Pj(b,x)) '&' Pj(c,x)) 'or'
(Pj(a '&' c,x) 'or' Pj(b '&' c,x)) by BVFUNC_1:def 7
.='not'( (Pj(a,x) 'or' Pj(b,x)) '&' Pj(c,x)) 'or'
((Pj(a,x) '&' Pj(c,x)) 'or' Pj(b '&' c,x)) by VALUAT_1:def 6
.='not'( Pj(c,x) '&' (Pj(a,x) 'or' Pj(b,x))) 'or'
((Pj(a,x) '&' Pj(c,x)) 'or' (Pj(b,x) '&' Pj(c,x))) by VALUAT_1:def 6
.='not'( (Pj(c,x) '&' Pj(a,x)) 'or' (Pj(c,x) '&' Pj(b,x))) 'or'
((Pj(a,x) '&' Pj(c,x)) 'or' (Pj(b,x) '&' Pj(c,x))) by BINARITH:22
.=((Pj(a,x) '&' Pj(c,x)) 'or' (Pj(b,x) '&' Pj(c,x))) 'or'
('not'( Pj(a,x) '&' Pj(c,x)) '&' 'not'( Pj(b,x) '&' Pj(c,x)))
by BINARITH:10
.=(((Pj(b,x) '&' Pj(c,x)) 'or' (Pj(a,x) '&' Pj(c,x))) 'or'
'not'( Pj(a,x) '&' Pj(c,x))) '&'
(((Pj(a,x) '&' Pj(c,x)) 'or' (Pj(b,x) '&' Pj(c,x))) 'or'
'not'( Pj(b,x) '&' Pj(c,x))) by BINARITH:23
.=((Pj(b,x) '&' Pj(c,x)) 'or' ((Pj(a,x) '&' Pj(c,x)) 'or'
'not'( Pj(a,x) '&' Pj(c,x)))) '&'
(((Pj(a,x) '&' Pj(c,x)) 'or' (Pj(b,x) '&' Pj(c,x))) 'or'
'not'( Pj(b,x) '&' Pj(c,x))) by BINARITH:20
.=((Pj(b,x) '&' Pj(c,x)) 'or' ((Pj(a,x) '&' Pj(c,x)) 'or'
'not'( Pj(a,x) '&' Pj(c,x)))) '&'
((Pj(a,x) '&' Pj(c,x)) 'or' ((Pj(b,x) '&' Pj(c,x)) 'or'
'not'( Pj(b,x) '&' Pj(c,x)))) by BINARITH:20
.=((Pj(b,x) '&' Pj(c,x)) 'or' TRUE) '&'
((Pj(a,x) '&' Pj(c,x)) 'or' ((Pj(b,x) '&' Pj(c,x)) 'or'
'not'( Pj(b,x) '&' Pj(c,x)))) by BINARITH:18
.=((Pj(b,x) '&' Pj(c,x)) 'or' TRUE) '&'
((Pj(a,x) '&' Pj(c,x)) 'or' TRUE) by BINARITH:18
.=TRUE '&' ((Pj(a,x) '&' Pj(c,x)) 'or' TRUE) by BINARITH:19
.=TRUE '&' TRUE by BINARITH:19
.=TRUE by MARGREL1:45;
hence thesis;
end;
hence thesis by BVFUNC_1:def 14;
end;

theorem for a,b being Element of Funcs(Y,BOOLEAN) holds
(a '&' b)=I_el(Y) implies (a 'or' b)=I_el(Y)
proof
let a,b be Element of Funcs(Y,BOOLEAN);
assume A1: (a '&' b)=I_el(Y);
for x being Element of Y holds
Pj(a 'or' b,x)=TRUE
proof
let x be Element of Y;
Pj(a '&' b,x)= TRUE by A1,BVFUNC_1:def 14;
then Pj(a,x) '&' Pj(b,x)=TRUE by VALUAT_1:def 6;
then Pj(a,x)=TRUE & Pj(b,x)=TRUE by MARGREL1:45;
then Pj(a 'or' b,x)
=TRUE 'or' TRUE by BVFUNC_1:def 7
.=TRUE by BINARITH:19;
hence thesis;
end;
hence thesis by BVFUNC_1:def 14;
end;

theorem for a,b,c being Element of Funcs(Y,BOOLEAN) holds
(a 'imp' b)=I_el(Y) implies (a 'or' c) 'imp' (b 'or' c)=I_el(Y)
proof
let a,b,c be Element of Funcs(Y,BOOLEAN);
assume A1: (a 'imp' b)=I_el(Y);
for x being Element of Y holds
Pj((a 'or' c) 'imp' (b 'or' c),x)=TRUE
proof
let x be Element of Y;
Pj(a 'imp' b,x)= TRUE by A1,BVFUNC_1:def 14;
then A2:'not' Pj(a,x) 'or' Pj(b,x) = TRUE by BVFUNC_1:def 11;
Pj((a 'or' c) 'imp' (b 'or' c),x)
='not' Pj(a 'or' c,x) 'or' Pj(b 'or' c,x) by BVFUNC_1:def 11
.='not'( Pj(a,x) 'or' Pj(c,x)) 'or' Pj(b 'or' c,x) by BVFUNC_1:def 7
.='not'( Pj(a,x) 'or' Pj(c,x)) 'or' (Pj(b,x) 'or' Pj(c,x)) by BVFUNC_1:def 7
.=(Pj(b,x) 'or' Pj(c,x)) 'or' ('not' Pj(a,x) '&' 'not' Pj(c,x))
by BINARITH:10
.=((Pj(c,x) 'or' Pj(b,x)) 'or' 'not' Pj(a,x)) '&'
((Pj(b,x) 'or' Pj(c,x)) 'or' 'not' Pj(c,x)) by BINARITH:23
.=(Pj(c,x) 'or' ('not' Pj(a,x) 'or' Pj(b,x))) '&'
((Pj(b,x) 'or' Pj(c,x)) 'or' 'not' Pj(c,x)) by BINARITH:20
.=TRUE '&' ((Pj(b,x) 'or' Pj(c,x)) 'or' 'not' Pj(c,x)) by A2,BINARITH:19
.=TRUE '&' (Pj(b,x) 'or' (Pj(c,x) 'or' 'not' Pj(c,x))) by BINARITH:20
.=TRUE '&' (Pj(b,x) 'or' TRUE) by BINARITH:18
.=TRUE '&' TRUE by BINARITH:19
.=TRUE by MARGREL1:45;
hence thesis;
end;
hence thesis by BVFUNC_1:def 14;
end;

theorem for a,b,c being Element of Funcs(Y,BOOLEAN) holds
(a 'imp' b)=I_el(Y) implies (a '&' c) 'imp' (b '&' c)=I_el(Y)
proof
let a,b,c be Element of Funcs(Y,BOOLEAN);
assume A1: (a 'imp' b)=I_el(Y);
for x being Element of Y holds
Pj((a '&' c) 'imp' (b '&' c),x)=TRUE
proof
let x be Element of Y;
Pj(a 'imp' b,x)= TRUE by A1,BVFUNC_1:def 14;
then A2:'not' Pj(a,x) 'or' Pj(b,x) = TRUE by BVFUNC_1:def 11;
Pj((a '&' c) 'imp' (b '&' c),x)
='not' Pj(a '&' c,x) 'or' Pj(b '&' c,x) by BVFUNC_1:def 11
.='not'( Pj(a,x) '&' Pj(c,x)) 'or' Pj(b '&' c,x) by VALUAT_1:def 6
.='not'( Pj(a,x) '&' Pj(c,x)) 'or' (Pj(b,x) '&' Pj(c,x)) by VALUAT_1:def 6
.=('not' Pj(a,x) 'or' 'not' Pj(c,x)) 'or' (Pj(b,x) '&' Pj(c,x))
by BINARITH:9
.=(('not' Pj(c,x) 'or' 'not' Pj(a,x)) 'or' Pj(b,x)) '&'
(('not' Pj(a,x) 'or' 'not' Pj(c,x)) 'or' Pj(c,x)) by BINARITH:23
.=('not' Pj(c,x) 'or' ('not' Pj(a,x) 'or' Pj(b,x))) '&'
(('not' Pj(a,x) 'or' 'not' Pj(c,x)) 'or' Pj(c,x)) by BINARITH:20
.=('not' Pj(c,x) 'or' ('not' Pj(a,x) 'or' Pj(b,x))) '&'
('not' Pj(a,x) 'or' ('not' Pj(c,x) 'or' Pj(c,x))) by BINARITH:20
.=('not' Pj(c,x) 'or' ('not' Pj(a,x) 'or' Pj(b,x))) '&'
('not' Pj(a,x) 'or' TRUE) by BINARITH:18
.=('not' Pj(c,x) 'or' TRUE) '&' TRUE by A2,BINARITH:19
.=TRUE '&' TRUE by BINARITH:19
.=TRUE by MARGREL1:45;
hence thesis;
end;
hence thesis by BVFUNC_1:def 14;
end;

theorem for a,b,c being Element of Funcs(Y,BOOLEAN) holds
(c 'imp' a)=I_el(Y) & (c 'imp' b)=I_el(Y) implies
c 'imp' (a '&' b)=I_el(Y)
proof
let a,b,c be Element of Funcs(Y,BOOLEAN);
assume A1:(c 'imp' a)=I_el(Y) & (c 'imp' b)=I_el(Y);
for x being Element of Y holds
Pj(c 'imp' (a '&' b),x)=TRUE
proof
let x be Element of Y;
Pj(c 'imp' a,x)= TRUE by A1,BVFUNC_1:def 14;
then A2:'not' Pj(c,x) 'or' Pj(a,x) = TRUE by BVFUNC_1:def 11;
Pj(c 'imp' b,x)= TRUE by A1,BVFUNC_1:def 14;
then A3:'not' Pj(c,x) 'or' Pj(b,x) = TRUE by BVFUNC_1:def 11;
Pj(c 'imp' (a '&' b),x)
='not' Pj(c,x) 'or' Pj(a '&' b,x) by BVFUNC_1:def 11
.='not' Pj(c,x) 'or' (Pj(a,x) '&' Pj(b,x)) by VALUAT_1:def 6
.=TRUE '&' TRUE by A2,A3,BINARITH:23
.=TRUE by MARGREL1:45;
hence thesis;
end;
hence thesis by BVFUNC_1:def 14;
end;

theorem for a,b,c being Element of Funcs(Y,BOOLEAN) holds
(a 'imp' c)=I_el(Y) & (b 'imp' c)=I_el(Y) implies
(a 'or' b) 'imp' c = I_el(Y)
proof
let a,b,c be Element of Funcs(Y,BOOLEAN);
assume A1:(a 'imp' c)=I_el(Y) & (b 'imp' c)=I_el(Y);
for x being Element of Y holds
Pj((a 'or' b) 'imp' c,x)=TRUE
proof
let x be Element of Y;
Pj(a 'imp' c,x)= TRUE by A1,BVFUNC_1:def 14;
then A2:'not' Pj(a,x) 'or' Pj(c,x) = TRUE by BVFUNC_1:def 11;
Pj(b 'imp' c,x)= TRUE by A1,BVFUNC_1:def 14;
then A3:'not' Pj(b,x) 'or' Pj(c,x) = TRUE by BVFUNC_1:def 11;
Pj((a 'or' b) 'imp' c,x)
='not' Pj(a 'or' b,x) 'or' Pj(c,x) by BVFUNC_1:def 11
.='not'( Pj(a,x) 'or' Pj(b,x)) 'or' Pj(c,x) by BVFUNC_1:def 7
.=Pj(c,x) 'or' ('not' Pj(a,x) '&' 'not' Pj(b,x)) by BINARITH:10
.=TRUE '&' TRUE by A2,A3,BINARITH:23
.=TRUE by MARGREL1:45;
hence thesis;
end;
hence thesis by BVFUNC_1:def 14;
end;

theorem for a,b being Element of Funcs(Y,BOOLEAN) holds
(a 'or' b)=I_el(Y) & 'not' a=I_el(Y) implies b=I_el(Y)
proof
let a,b be Element of Funcs(Y,BOOLEAN);
assume A1:(a 'or' b)=I_el(Y) & 'not' a=I_el(Y);
for x being Element of Y holds
Pj(b,x)=TRUE
proof
let x be Element of Y;
Pj(a 'or' b,x)= TRUE by A1,BVFUNC_1:def 14;
then A2:Pj(a,x) 'or' Pj(b,x) = TRUE by BVFUNC_1:def 7;
Pj('not' a,x)= TRUE by A1,BVFUNC_1:def 14;
then 'not' Pj(a,x) = TRUE by VALUAT_1:def 5;
then Pj(a,x) = FALSE by MARGREL1:41;
hence thesis by A2,BINARITH:7;
end;
hence thesis by BVFUNC_1:def 14;
end;

theorem for a,b,c,d being Element of Funcs(Y,BOOLEAN) holds
(a 'imp' b)=I_el(Y) & (c 'imp' d)=I_el(Y) implies
(a '&' c) 'imp' (b '&' d)=I_el(Y)
proof
let a,b,c,d be Element of Funcs(Y,BOOLEAN);
assume A1:(a 'imp' b)=I_el(Y) & (c 'imp' d)=I_el(Y);
for x being Element of Y holds
Pj((a '&' c) 'imp' (b '&' d),x)=TRUE
proof
let x be Element of Y;
Pj(a 'imp' b,x)= TRUE by A1,BVFUNC_1:def 14;
then A2:'not' Pj(a,x) 'or' Pj(b,x) = TRUE by BVFUNC_1:def 11;
Pj(c 'imp' d,x)= TRUE by A1,BVFUNC_1:def 14;
then A3:'not' Pj(c,x) 'or' Pj(d,x) = TRUE by BVFUNC_1:def 11;
Pj((a '&' c) 'imp' (b '&' d),x)
='not' Pj(a '&' c,x) 'or' Pj(b '&' d,x) by BVFUNC_1:def 11
.='not'( Pj(a,x) '&' Pj(c,x)) 'or' Pj(b '&' d,x) by VALUAT_1:def 6
.='not'( Pj(a,x) '&' Pj(c,x)) 'or' (Pj(b,x) '&' Pj(d,x)) by VALUAT_1:def 6
.=('not' Pj(a,x) 'or' 'not' Pj(c,x)) 'or' (Pj(b,x) '&' Pj(d,x))
by BINARITH:9
.=(('not' Pj(c,x) 'or' 'not' Pj(a,x)) 'or' Pj(b,x)) '&'
(('not' Pj(a,x) 'or' 'not' Pj(c,x)) 'or' Pj(d,x)) by BINARITH:23
.=('not' Pj(c,x) 'or' ('not' Pj(a,x) 'or' Pj(b,x))) '&'
(('not' Pj(a,x) 'or' 'not' Pj(c,x)) 'or' Pj(d,x)) by BINARITH:20
.=('not' Pj(c,x) 'or' TRUE) '&' ('not' Pj(a,x) 'or' TRUE)
by A2,A3,BINARITH:20
.=TRUE '&' ('not' Pj(a,x) 'or' TRUE) by BINARITH:19
.=TRUE '&' TRUE by BINARITH:19
.=TRUE by MARGREL1:45;
hence thesis;
end;
hence thesis by BVFUNC_1:def 14;
end;

theorem for a,b,c,d being Element of Funcs(Y,BOOLEAN) holds
(a 'imp' b)=I_el(Y) & (c 'imp' d)=I_el(Y) implies
(a 'or' c) 'imp' (b 'or' d) =I_el(Y)
proof
let a,b,c,d be Element of Funcs(Y,BOOLEAN);
assume A1:(a 'imp' b)=I_el(Y) & (c 'imp' d)=I_el(Y);
for x being Element of Y holds
Pj((a 'or' c) 'imp' (b 'or' d),x)=TRUE
proof
let x be Element of Y;
Pj(a 'imp' b,x)= TRUE by A1,BVFUNC_1:def 14;
then A2:'not' Pj(a,x) 'or' Pj(b,x) = TRUE by BVFUNC_1:def 11;
Pj(c 'imp' d,x)= TRUE by A1,BVFUNC_1:def 14;
then A3:'not' Pj(c,x) 'or' Pj(d,x) = TRUE by BVFUNC_1:def 11;
Pj((a 'or' c) 'imp' (b 'or' d),x)
='not' Pj(a 'or' c,x) 'or' Pj(b 'or' d,x) by BVFUNC_1:def 11
.='not'( Pj(a,x) 'or' Pj(c,x)) 'or' Pj(b 'or' d,x) by BVFUNC_1:def 7
.='not'( Pj(a,x) 'or' Pj(c,x)) 'or' (Pj(b,x) 'or' Pj(d,x)) by BVFUNC_1:def 7
.=(Pj(b,x) 'or' Pj(d,x)) 'or' ('not' Pj(a,x) '&' 'not' Pj(c,x))
by BINARITH:10
.=((Pj(d,x) 'or' Pj(b,x)) 'or' 'not' Pj(a,x)) '&'
((Pj(b,x) 'or' Pj(d,x)) 'or' 'not' Pj(c,x)) by BINARITH:23
.=(Pj(d,x) 'or' (Pj(b,x) 'or' 'not' Pj(a,x))) '&'
((Pj(b,x) 'or' Pj(d,x)) 'or' 'not' Pj(c,x)) by BINARITH:20
.=(Pj(d,x) 'or' TRUE) '&' (Pj(b,x) 'or' TRUE) by A2,A3,BINARITH:20
.=TRUE '&' (Pj(b,x) 'or' TRUE) by BINARITH:19
.=TRUE '&' TRUE by BINARITH:19
.=TRUE by MARGREL1:45;
hence thesis;
end;
hence thesis by BVFUNC_1:def 14;
end;

theorem for a,b being Element of Funcs(Y,BOOLEAN) holds
(a '&' 'not' b) 'imp' 'not' a=I_el(Y) implies (a 'imp' b)=I_el(Y)
proof
let a,b be Element of Funcs(Y,BOOLEAN);
assume A1: (a '&' 'not' b) 'imp' 'not' a=I_el(Y);
for x being Element of Y holds
Pj(a 'imp' b,x)=TRUE
proof
let x be Element of Y;
Pj((a '&' 'not' b) 'imp' 'not' a,x)=TRUE by A1,BVFUNC_1:def 14;
then 'not' Pj(a '&' 'not' b,x) 'or' Pj('not' a,x) = TRUE by BVFUNC_1:def 11
;
then 'not'( Pj(a,x) '&' Pj('not' b,x)) 'or' Pj('not'
a,x)=TRUE by VALUAT_1:def 6;
then ('not' Pj(a,x) 'or' 'not' Pj('not' b,x)) 'or' Pj('not'
a,x)=TRUE by BINARITH:9;
then ('not' Pj(a,x) 'or' 'not' 'not' Pj(b,x)) 'or' Pj('not'
a,x)=TRUE by VALUAT_1:def 5;
then ('not' Pj(a,x) 'or' 'not' 'not' Pj(b,x)) 'or' 'not'
Pj(a,x)=TRUE by VALUAT_1:def 5;
then ('not' Pj(a,x) 'or' Pj(b,x)) 'or' 'not' Pj(a,x)=TRUE by MARGREL1:40;
then Pj(b,x) 'or' ('not' Pj(a,x) 'or' 'not' Pj(a,x))=TRUE by BINARITH:20;
then Pj(b,x) 'or' 'not' Pj(a,x)=TRUE by BINARITH:21;
hence thesis by BVFUNC_1:def 11;
end;
hence thesis by BVFUNC_1:def 14;
end;

canceled;

theorem for a,b being Element of Funcs(Y,BOOLEAN) holds
a 'imp' 'not' b=I_el(Y) implies b 'imp' 'not' a=I_el(Y)
proof
let a,b be Element of Funcs(Y,BOOLEAN);
assume A1:a 'imp' 'not' b=I_el(Y);
for x being Element of Y holds Pj(b 'imp' 'not' a,x)=TRUE
proof
let x be Element of Y;
Pj(a 'imp' 'not' b,x)=TRUE by A1,BVFUNC_1:def 14;
then ('not' Pj(a,x)) 'or' Pj('not' b,x)=TRUE by BVFUNC_1:def 11;
then A2:'not' Pj(a,x) 'or' 'not' Pj(b,x)=TRUE by VALUAT_1:def 5;
Pj(b 'imp' 'not' a,x)
=('not' Pj(b,x)) 'or' Pj('not' a,x) by BVFUNC_1:def 11
.=TRUE by A2,VALUAT_1:def 5;
hence thesis;
end;
hence thesis by BVFUNC_1:def 14;
end;

theorem for a,b being Element of Funcs(Y,BOOLEAN) holds
'not' a 'imp' b=I_el(Y) implies 'not' b 'imp' a=I_el(Y)
proof
let a,b be Element of Funcs(Y,BOOLEAN);
assume A1:'not' a 'imp' b=I_el(Y);
for x being Element of Y holds Pj('not' b 'imp' a,x)=TRUE
proof
let x be Element of Y;
Pj('not' a 'imp' b,x)=TRUE by A1,BVFUNC_1:def 14;
then ('not' Pj('not' a,x)) 'or' Pj(b,x)=TRUE by BVFUNC_1:def 11;
then 'not' 'not' Pj(a,x) 'or' Pj(b,x)=TRUE by VALUAT_1:def 5;
then A2:Pj(a,x) 'or' Pj(b,x)=TRUE by MARGREL1:40;
Pj('not' b 'imp' a,x)
=('not' Pj('not' b,x)) 'or' Pj(a,x) by BVFUNC_1:def 11
.=('not' 'not' Pj(b,x)) 'or' Pj(a,x) by VALUAT_1:def 5
.=TRUE by A2,MARGREL1:40;
hence thesis;
end;
hence thesis by BVFUNC_1:def 14;
end;

theorem for a,b being Element of Funcs(Y,BOOLEAN) holds
a 'imp' (a 'or' b)=I_el(Y)
proof
let a,b be Element of Funcs(Y,BOOLEAN);
for x being Element of Y holds Pj(a 'imp' (a 'or' b),x)=TRUE
proof
let x be Element of Y;
Pj(a 'imp' (a 'or' b),x)
='not' Pj(a,x) 'or' Pj(a 'or' b,x) by BVFUNC_1:def 11
.='not' Pj(a,x) 'or' (Pj(a,x) 'or' Pj(b,x)) by BVFUNC_1:def 7
.=('not' Pj(a,x) 'or' Pj(a,x)) 'or' Pj(b,x) by BINARITH:20
.=TRUE 'or' Pj(b,x) by BINARITH:18
.=TRUE by BINARITH:19;
hence thesis;
end;
hence thesis by BVFUNC_1:def 14;
end;

theorem for a,b being Element of Funcs(Y,BOOLEAN) holds
(a 'or' b) 'imp' ('not' a 'imp' b)=I_el(Y)
proof
let a,b be Element of Funcs(Y,BOOLEAN);
for x being Element of Y holds Pj((a 'or' b) 'imp' ('not' a 'imp' b),x)=
TRUE
proof
let x be Element of Y;
Pj((a 'or' b) 'imp' ('not' a 'imp' b),x)
='not' Pj(a 'or' b,x) 'or' Pj('not' a 'imp' b,x) by BVFUNC_1:def 11
.='not'( Pj(a,x) 'or' Pj(b,x)) 'or' Pj('not' a 'imp' b,x) by BVFUNC_1:def 7
.='not'( Pj(a,x) 'or' Pj(b,x)) 'or' ('not' Pj('not'
a,x) 'or' Pj(b,x)) by BVFUNC_1:def 11
.=('not' Pj(a,x) '&' 'not' Pj(b,x)) 'or' ('not' Pj('not'
a,x) 'or' Pj(b,x)) by BINARITH:10
.=('not' Pj(a,x) '&' 'not' Pj(b,x)) 'or' ('not' 'not'
Pj(a,x) 'or' Pj(b,x)) by VALUAT_1:def 5
.=(Pj(a,x) 'or' Pj(b,x)) 'or' ('not' Pj(a,x) '&' 'not' Pj(b,x))
by MARGREL1:40
.=((Pj(a,x) 'or' Pj(b,x)) 'or' 'not' Pj(a,x)) '&'
((Pj(a,x) 'or' Pj(b,x)) 'or' 'not' Pj(b,x)) by BINARITH:23
.=((Pj(a,x) 'or' 'not' Pj(a,x)) 'or' Pj(b,x)) '&'
((Pj(a,x) 'or' Pj(b,x)) 'or' 'not' Pj(b,x)) by BINARITH:20
.=((Pj(a,x) 'or' 'not' Pj(a,x)) 'or' Pj(b,x)) '&'
(Pj(a,x) 'or' (Pj(b,x) 'or' 'not' Pj(b,x))) by BINARITH:20
.=(TRUE 'or' Pj(b,x)) '&'
(Pj(a,x) 'or' (Pj(b,x) 'or' 'not' Pj(b,x))) by BINARITH:18
.=(TRUE 'or' Pj(b,x)) '&' (Pj(a,x) 'or' TRUE) by BINARITH:18
.=TRUE '&' (Pj(a,x) 'or' TRUE) by BINARITH:19
.=TRUE '&' TRUE by BINARITH:19
.=TRUE by MARGREL1:45;
hence thesis;
end;
hence thesis by BVFUNC_1:def 14;
end;

theorem for a,b being Element of Funcs(Y,BOOLEAN) holds
'not'( a 'or' b) 'imp' ('not' a '&' 'not' b)=I_el(Y)
proof
let a,b be Element of Funcs(Y,BOOLEAN);
for x being Element of Y holds Pj('not'( a 'or' b) 'imp' ('not' a '&' 'not'
b),x)=TRUE
proof
let x be Element of Y;
Pj('not'( a 'or' b) 'imp' ('not' a '&' 'not' b),x)
='not' Pj('not'( a 'or' b),x) 'or' Pj('not' a '&' 'not' b,x)
by BVFUNC_1:def 11
.='not' 'not' Pj(a 'or' b,x) 'or' Pj('not' a '&' 'not' b,x)
by VALUAT_1:def 5
.=Pj(a 'or' b,x) 'or' Pj('not' a '&' 'not' b,x) by MARGREL1:40
.=(Pj(a,x) 'or' Pj(b,x)) 'or' Pj('not' a '&' 'not' b,x) by BVFUNC_1:def 7
.=(Pj(a,x) 'or' Pj(b,x)) 'or' (Pj('not' a,x) '&' Pj('not'
b,x)) by VALUAT_1:def 6
.=(Pj(a,x) 'or' Pj(b,x)) 'or' ('not' Pj(a,x) '&' Pj('not'
b,x)) by VALUAT_1:def 5
.=(Pj(a,x) 'or' Pj(b,x)) 'or' ('not' Pj(a,x) '&' 'not'
Pj(b,x)) by VALUAT_1:def 5
.=((Pj(a,x) 'or' Pj(b,x)) 'or' 'not' Pj(a,x)) '&'
((Pj(a,x) 'or' Pj(b,x)) 'or' 'not' Pj(b,x)) by BINARITH:23
.=((Pj(a,x) 'or' 'not' Pj(a,x)) 'or' Pj(b,x)) '&'
((Pj(a,x) 'or' Pj(b,x)) 'or' 'not' Pj(b,x)) by BINARITH:20
.=((Pj(a,x) 'or' 'not' Pj(a,x)) 'or' Pj(b,x)) '&'
(Pj(a,x) 'or' (Pj(b,x) 'or' 'not' Pj(b,x))) by BINARITH:20
.=(TRUE 'or' Pj(b,x)) '&'
(Pj(a,x) 'or' (Pj(b,x) 'or' 'not' Pj(b,x))) by BINARITH:18
.=(TRUE 'or' Pj(b,x)) '&' (Pj(a,x) 'or' TRUE) by BINARITH:18
.=TRUE '&' (Pj(a,x) 'or' TRUE) by BINARITH:19
.=TRUE '&' TRUE by BINARITH:19
.=TRUE by MARGREL1:45;
hence thesis;
end;
hence thesis by BVFUNC_1:def 14;
end;

theorem for a,b being Element of Funcs(Y,BOOLEAN) holds
('not' a '&' 'not' b) 'imp' 'not'( a 'or' b)=I_el(Y)
proof
let a,b be Element of Funcs(Y,BOOLEAN);
for x being Element of Y holds Pj(('not' a '&' 'not' b) 'imp' 'not'
(a 'or' b),x)=TRUE
proof
let x be Element of Y;
Pj(('not' a '&' 'not' b) 'imp' 'not'( a 'or' b),x)
='not' Pj('not' a '&' 'not' b,x) 'or' Pj('not'( a 'or' b),x)
by BVFUNC_1:def 11
.='not' Pj('not' a '&' 'not' b,x) 'or' 'not' Pj(a 'or' b,x)
by VALUAT_1:def 5
.='not'( Pj('not' a,x) '&' Pj('not' b,x)) 'or' 'not'
Pj(a 'or' b,x) by VALUAT_1:def 6
.='not'( 'not' Pj(a,x) '&' Pj('not' b,x)) 'or' 'not'
Pj(a 'or' b,x) by VALUAT_1:def 5
.='not'( 'not' Pj(a,x) '&' 'not' Pj(b,x)) 'or' 'not'
Pj(a 'or' b,x) by VALUAT_1:def 5
.=('not' 'not' Pj(a,x) 'or' 'not' 'not' Pj(b,x)) 'or' 'not'
Pj(a 'or' b,x) by BINARITH:9
.=(Pj(a,x) 'or' 'not' 'not' Pj(b,x)) 'or' 'not' Pj(a 'or' b,x)
by MARGREL1:40
.=(Pj(a,x) 'or' Pj(b,x)) 'or' 'not' Pj(a 'or' b,x) by MARGREL1:40
.=(Pj(a,x) 'or' Pj(b,x)) 'or' 'not'( Pj(a,x) 'or' Pj(b,x))
by BVFUNC_1:def 7
.=(Pj(a,x) 'or' Pj(b,x)) 'or' ('not' Pj(a,x) '&' 'not' Pj(b,x))
by BINARITH:10
.=((Pj(a,x) 'or' Pj(b,x)) 'or' 'not' Pj(a,x)) '&'
((Pj(a,x) 'or' Pj(b,x)) 'or' 'not' Pj(b,x)) by BINARITH:23
.=((Pj(a,x) 'or' 'not' Pj(a,x)) 'or' Pj(b,x)) '&'
((Pj(a,x) 'or' Pj(b,x)) 'or' 'not' Pj(b,x)) by BINARITH:20
.=((Pj(a,x) 'or' 'not' Pj(a,x)) 'or' Pj(b,x)) '&'
(Pj(a,x) 'or' (Pj(b,x) 'or' 'not' Pj(b,x))) by BINARITH:20
.=(TRUE 'or' Pj(b,x)) '&'
(Pj(a,x) 'or' (Pj(b,x) 'or' 'not' Pj(b,x))) by BINARITH:18
.=(TRUE 'or' Pj(b,x)) '&' (Pj(a,x) 'or' TRUE) by BINARITH:18
.=TRUE '&' (Pj(a,x) 'or' TRUE) by BINARITH:19
.=TRUE '&' TRUE by BINARITH:19
.=TRUE by MARGREL1:45;
hence thesis;
end;
hence thesis by BVFUNC_1:def 14;
end;

theorem for a,b being Element of Funcs(Y,BOOLEAN) holds
'not'( a 'or' b) 'imp' 'not' a=I_el(Y)
proof
let a,b be Element of Funcs(Y,BOOLEAN);
for x being Element of Y holds Pj('not'( a 'or' b) 'imp' 'not' a,x)=TRUE
proof
let x be Element of Y;
Pj('not'( a 'or' b) 'imp' 'not' a,x)
='not' Pj('not'( a 'or' b),x) 'or' Pj('not' a,x) by BVFUNC_1:def 11
.='not' 'not' Pj(a 'or' b,x) 'or' Pj('not' a,x) by VALUAT_1:def 5
.='not' 'not' Pj(a 'or' b,x) 'or' 'not' Pj(a,x) by VALUAT_1:def 5
.=Pj(a 'or' b,x) 'or' 'not' Pj(a,x) by MARGREL1:40
.=(Pj(a,x) 'or' Pj(b,x)) 'or' 'not' Pj(a,x) by BVFUNC_1:def 7
.=(Pj(a,x) 'or' 'not' Pj(a,x)) 'or' Pj(b,x) by BINARITH:20
.=TRUE 'or' Pj(b,x) by BINARITH:18
.=TRUE by BINARITH:19;
hence thesis;
end;
hence thesis by BVFUNC_1:def 14;
end;

theorem for a being Element of Funcs(Y,BOOLEAN) holds
(a 'or' a) 'imp' a=I_el(Y)
proof
let a be Element of Funcs(Y,BOOLEAN);
for x being Element of Y holds Pj((a 'or' a) 'imp' a,x)=TRUE
proof
let x be Element of Y;
Pj((a 'or' a) 'imp' a,x)
='not' Pj(a 'or' a,x) 'or' Pj(a,x) by BVFUNC_1:def 11
.='not'( Pj(a,x) 'or' Pj(a,x)) 'or' Pj(a,x) by BVFUNC_1:def 7
.=('not' Pj(a,x) '&' 'not' Pj(a,x)) 'or' Pj(a,x) by BINARITH:10
.='not' Pj(a,x) 'or' Pj(a,x) by BINARITH:16
.=TRUE by BINARITH:18;
hence thesis;
end;
hence thesis by BVFUNC_1:def 14;
end;

theorem for a,b being Element of Funcs(Y,BOOLEAN) holds
(a '&' 'not' a) 'imp' b=I_el(Y)
proof
let a,b be Element of Funcs(Y,BOOLEAN);
for x being Element of Y holds Pj((a '&' 'not' a) 'imp' b,x)=TRUE
proof
let x be Element of Y;
Pj((a '&' 'not' a) 'imp' b,x)
='not' Pj(a '&' 'not' a,x) 'or' Pj(b,x) by BVFUNC_1:def 11
.='not'( Pj(a,x) '&' Pj('not' a,x)) 'or' Pj(b,x) by VALUAT_1:def 6
.='not'( Pj(a,x) '&' 'not' Pj(a,x)) 'or' Pj(b,x) by VALUAT_1:def 5
.=('not' Pj(a,x) 'or' 'not' 'not' Pj(a,x)) 'or' Pj(b,x) by BINARITH:9
.=TRUE 'or' Pj(b,x) by BINARITH:18
.=TRUE by BINARITH:19;
hence thesis;
end;
hence thesis by BVFUNC_1:def 14;
end;

theorem for a,b being Element of Funcs(Y,BOOLEAN) holds
(a 'imp' b) 'imp' ('not' a 'or' b)=I_el(Y)
proof
let a,b be Element of Funcs(Y,BOOLEAN);
for x being Element of Y holds Pj((a 'imp' b) 'imp' ('not' a 'or' b),x)=
TRUE
proof
let x be Element of Y;
Pj((a 'imp' b) 'imp' ('not' a 'or' b),x)
='not' Pj(a 'imp' b,x) 'or' Pj('not' a 'or' b,x) by BVFUNC_1:def 11
.='not'( 'not' Pj(a,x) 'or' Pj(b,x)) 'or' Pj('not'
a 'or' b,x) by BVFUNC_1:def 11
.='not'( 'not' Pj(a,x) 'or' Pj(b,x)) 'or' (Pj('not'
a,x) 'or' Pj(b,x)) by BVFUNC_1:def 7
.=('not' 'not' Pj(a,x) '&' 'not' Pj(b,x)) 'or' (Pj('not'
a,x) 'or' Pj(b,x)) by BINARITH:10
.=(Pj(a,x) '&' 'not' Pj(b,x)) 'or' (Pj('not' a,x) 'or' Pj(b,x))
by MARGREL1:40
.=('not' Pj(a,x) 'or' Pj(b,x)) 'or' (Pj(a,x) '&' 'not'
Pj(b,x)) by VALUAT_1:def 5
.=(('not' Pj(a,x) 'or' Pj(b,x)) 'or' Pj(a,x)) '&'
(('not' Pj(a,x) 'or' Pj(b,x)) 'or' 'not' Pj(b,x)) by BINARITH:23
.=(('not' Pj(a,x) 'or' Pj(a,x)) 'or' Pj(b,x)) '&'
(('not' Pj(a,x) 'or' Pj(b,x)) 'or' 'not' Pj(b,x)) by BINARITH:20
.=(('not' Pj(a,x) 'or' Pj(a,x)) 'or' Pj(b,x)) '&'
('not' Pj(a,x) 'or' (Pj(b,x) 'or' 'not' Pj(b,x))) by BINARITH:20
.=(TRUE 'or' Pj(b,x)) '&'
('not' Pj(a,x) 'or' (Pj(b,x) 'or' 'not' Pj(b,x))) by BINARITH:18
.=(TRUE 'or' Pj(b,x)) '&' ('not' Pj(a,x) 'or' TRUE) by BINARITH:18
.=TRUE '&' ('not' Pj(a,x) 'or' TRUE) by BINARITH:19
.=TRUE '&' TRUE by BINARITH:19
.=TRUE by MARGREL1:45;
hence thesis;
end;
hence thesis by BVFUNC_1:def 14;
end;

theorem for a,b being Element of Funcs(Y,BOOLEAN) holds
(a '&' b) 'imp' 'not'( a 'imp' 'not' b)=I_el(Y)
proof
let a,b be Element of Funcs(Y,BOOLEAN);
for x being Element of Y holds Pj((a '&' b) 'imp' 'not'( a 'imp' 'not'
b),x)=TRUE
proof
let x be Element of Y;
Pj((a '&' b) 'imp' 'not'( a 'imp' 'not' b),x)
='not' Pj(a '&' b,x) 'or' Pj('not'( a 'imp' 'not' b),x) by BVFUNC_1:def 11
.='not'( Pj(a,x) '&' Pj(b,x)) 'or' Pj('not'( a 'imp' 'not'
b),x) by VALUAT_1:def 6
.='not'( Pj(a,x) '&' Pj(b,x)) 'or' 'not' Pj(a 'imp' 'not'
b,x) by VALUAT_1:def 5
.=('not' Pj(a,x) 'or' 'not' Pj(b,x)) 'or' 'not' Pj(a 'imp' 'not'
b,x) by BINARITH:9
.=('not' Pj(a,x) 'or' 'not' Pj(b,x)) 'or' 'not'( 'not' Pj(a,x) 'or' Pj('not'
b,x)) by BVFUNC_1:def 11
.=('not' Pj(a,x) 'or' 'not' Pj(b,x)) 'or' 'not'( 'not' Pj(a,x) 'or' 'not'
Pj(b,x)) by VALUAT_1:def 5
.=('not' Pj(a,x) 'or' 'not' Pj(b,x)) 'or' ('not' 'not'
Pj(a,x) '&' 'not' 'not'
Pj(b,x)) by BINARITH:10
.=('not' Pj(a,x) 'or' 'not' Pj(b,x)) 'or' (Pj(a,x) '&' 'not' 'not'
Pj(b,x)) by MARGREL1:40
.=('not' Pj(a,x) 'or' 'not' Pj(b,x)) 'or' (Pj(a,x) '&' Pj(b,x))
by MARGREL1:40
.=(('not' Pj(a,x) 'or' 'not' Pj(b,x)) 'or' Pj(a,x)) '&'
(('not' Pj(a,x) 'or' 'not' Pj(b,x)) 'or' Pj(b,x)) by BINARITH:23
.=(('not' Pj(a,x) 'or' Pj(a,x)) 'or' 'not' Pj(b,x)) '&'
(('not' Pj(a,x) 'or' 'not' Pj(b,x)) 'or' Pj(b,x)) by BINARITH:20
.=(('not' Pj(a,x) 'or' Pj(a,x)) 'or' 'not' Pj(b,x)) '&'
('not' Pj(a,x) 'or' ('not' Pj(b,x) 'or' Pj(b,x))) by BINARITH:20
.=(TRUE 'or' 'not' Pj(b,x)) '&'
('not' Pj(a,x) 'or' ('not' Pj(b,x) 'or' Pj(b,x))) by BINARITH:18
.=(TRUE 'or' 'not' Pj(b,x)) '&'
('not' Pj(a,x) 'or' TRUE) by BINARITH:18
.=TRUE '&' ('not' Pj(a,x) 'or' TRUE) by BINARITH:19
.=TRUE '&' TRUE by BINARITH:19
.=TRUE by MARGREL1:45;
hence thesis;
end;
hence thesis by BVFUNC_1:def 14;
end;

theorem for a,b being Element of Funcs(Y,BOOLEAN) holds
'not'( a 'imp' 'not' b) 'imp' (a '&' b)=I_el(Y)
proof
let a,b be Element of Funcs(Y,BOOLEAN);
for x being Element of Y holds Pj('not'( a 'imp' 'not'
b) 'imp' (a '&' b),x)=TRUE
proof
let x be Element of Y;
Pj('not'( a 'imp' 'not' b) 'imp' (a '&' b),x)
='not' Pj('not'( a 'imp' 'not' b),x) 'or' Pj(a '&' b,x) by BVFUNC_1:def 11
.='not' 'not' Pj(a 'imp' 'not' b,x) 'or' Pj(a '&' b,x) by VALUAT_1:def 5
.='not' 'not' Pj(a 'imp' 'not'
b,x) 'or' (Pj(a,x) '&' Pj(b,x)) by VALUAT_1:def 6
.='not' 'not'( 'not' Pj(a,x) 'or' Pj('not'
b,x)) 'or' (Pj(a,x) '&' Pj(b,x)) by BVFUNC_1:def 11
.=('not' Pj(a,x) 'or' Pj('not' b,x)) 'or' (Pj(a,x) '&' Pj(b,x))
by MARGREL1:40
.=('not' Pj(a,x) 'or' 'not'
Pj(b,x)) 'or' (Pj(a,x) '&' Pj(b,x)) by VALUAT_1:def 5
.=(('not' Pj(a,x) 'or' 'not' Pj(b,x)) 'or' Pj(a,x)) '&'
(('not' Pj(a,x) 'or' 'not' Pj(b,x)) 'or' Pj(b,x)) by BINARITH:23
.=(('not' Pj(a,x) 'or' Pj(a,x)) 'or' 'not' Pj(b,x)) '&'
(('not' Pj(a,x) 'or' 'not' Pj(b,x)) 'or' Pj(b,x)) by BINARITH:20
.=(('not' Pj(a,x) 'or' Pj(a,x)) 'or' 'not' Pj(b,x)) '&'
('not' Pj(a,x) 'or' ('not' Pj(b,x) 'or' Pj(b,x))) by BINARITH:20
.=(TRUE 'or' 'not' Pj(b,x)) '&'
('not' Pj(a,x) 'or' ('not' Pj(b,x) 'or' Pj(b,x))) by BINARITH:18
.=(TRUE 'or' 'not' Pj(b,x)) '&'
('not' Pj(a,x) 'or' TRUE) by BINARITH:18
.=TRUE '&' ('not' Pj(a,x) 'or' TRUE) by BINARITH:19
.=TRUE '&' TRUE by BINARITH:19
.=TRUE by MARGREL1:45;
hence thesis;
end;
hence thesis by BVFUNC_1:def 14;
end;

theorem for a,b being Element of Funcs(Y,BOOLEAN) holds
'not'( a '&' b) 'imp' ('not' a 'or' 'not' b)=I_el(Y)
proof
let a,b be Element of Funcs(Y,BOOLEAN);
for x being Element of Y holds Pj('not'( a '&' b) 'imp' ('not' a 'or' 'not'
b),x)=TRUE
proof
let x be Element of Y;
Pj('not'( a '&' b) 'imp' ('not' a 'or' 'not' b),x)
='not' Pj('not'( a '&' b),x) 'or' Pj('not' a 'or' 'not' b,x)
by BVFUNC_1:def 11
.='not' 'not' Pj(a '&' b,x) 'or' Pj('not' a 'or' 'not' b,x)
by VALUAT_1:def 5
.=Pj(a '&' b,x) 'or' Pj('not' a 'or' 'not' b,x) by MARGREL1:40
.=(Pj(a,x) '&' Pj(b,x)) 'or' Pj('not' a 'or' 'not' b,x)
by VALUAT_1:def 6
.=(Pj('not' a,x) 'or' Pj('not'
b,x)) 'or' (Pj(a,x) '&' Pj(b,x)) by BVFUNC_1:def 7
.=('not' Pj(a,x) 'or' Pj('not'
b,x)) 'or' (Pj(a,x) '&' Pj(b,x)) by VALUAT_1:def 5
.=('not' Pj(a,x) 'or' 'not'
Pj(b,x)) 'or' (Pj(a,x) '&' Pj(b,x)) by VALUAT_1:def 5
.=(('not' Pj(a,x) 'or' 'not' Pj(b,x)) 'or' Pj(a,x)) '&'
(('not' Pj(a,x) 'or' 'not' Pj(b,x)) 'or' Pj(b,x)) by BINARITH:23
.=(('not' Pj(a,x) 'or' Pj(a,x)) 'or' 'not' Pj(b,x)) '&'
(('not' Pj(a,x) 'or' 'not' Pj(b,x)) 'or' Pj(b,x)) by BINARITH:20
.=(('not' Pj(a,x) 'or' Pj(a,x)) 'or' 'not' Pj(b,x)) '&'
('not' Pj(a,x) 'or' ('not' Pj(b,x) 'or' Pj(b,x))) by BINARITH:20
.=(TRUE 'or' 'not' Pj(b,x)) '&'
('not' Pj(a,x) 'or' ('not' Pj(b,x) 'or' Pj(b,x))) by BINARITH:18
.=(TRUE 'or' 'not' Pj(b,x)) '&' ('not' Pj(a,x) 'or' TRUE) by BINARITH:18
.=TRUE '&' ('not' Pj(a,x) 'or' TRUE) by BINARITH:19
.=TRUE '&' TRUE by BINARITH:19
.=TRUE by MARGREL1:45;
hence thesis;
end;
hence thesis by BVFUNC_1:def 14;
end;

theorem for a,b being Element of Funcs(Y,BOOLEAN) holds
('not' a 'or' 'not' b) 'imp' 'not'( a '&' b)=I_el(Y)
proof
let a,b be Element of Funcs(Y,BOOLEAN);
for x being Element of Y holds Pj(('not' a 'or' 'not' b) 'imp' 'not'
(a '&' b),x)=TRUE
proof
let x be Element of Y;
Pj(('not' a 'or' 'not' b) 'imp' 'not'( a '&' b),x)
='not' Pj('not' a 'or' 'not' b,x) 'or' Pj('not'( a '&' b),x)
by BVFUNC_1:def 11
.='not' Pj('not' a 'or' 'not' b,x) 'or' 'not' Pj(a '&' b,x)
by VALUAT_1:def 5
.='not'( Pj('not' a,x) 'or' Pj('not' b,x)) 'or' 'not'
Pj(a '&' b,x) by BVFUNC_1:def 7
.='not'( 'not' Pj(a,x) 'or' Pj('not' b,x)) 'or' 'not'
Pj(a '&' b,x) by VALUAT_1:def 5
.='not'( 'not' Pj(a,x) 'or' 'not' Pj(b,x)) 'or' 'not'
Pj(a '&' b,x) by VALUAT_1:def 5
.='not'( 'not' Pj(a,x) 'or' 'not' Pj(b,x)) 'or' 'not'
(Pj(a,x) '&' Pj(b,x)) by VALUAT_1:def 6
.=('not' 'not' Pj(a,x) '&' 'not' 'not' Pj(b,x)) 'or' 'not'
(Pj(a,x) '&' Pj(b,x)) by BINARITH:10
.=('not' 'not' Pj(a,x) '&' 'not' 'not' Pj(b,x)) 'or'
('not' Pj(a,x) 'or' 'not'
Pj(b,x)) by BINARITH:9
.=(Pj(a,x) '&' 'not' 'not' Pj(b,x)) 'or' ('not' Pj(a,x) 'or' 'not'
Pj(b,x)) by MARGREL1:40
.=('not' Pj(a,x) 'or' 'not' Pj(b,x)) 'or' (Pj(a,x) '&' Pj(b,x))
by MARGREL1:40
.=(('not' Pj(a,x) 'or' 'not' Pj(b,x)) 'or' Pj(a,x)) '&'
(('not' Pj(a,x) 'or' 'not' Pj(b,x)) 'or' Pj(b,x)) by BINARITH:23
.=(('not' Pj(a,x) 'or' Pj(a,x)) 'or' 'not' Pj(b,x)) '&'
(('not' Pj(a,x) 'or' 'not' Pj(b,x)) 'or' Pj(b,x)) by BINARITH:20
.=(('not' Pj(a,x) 'or' Pj(a,x)) 'or' 'not' Pj(b,x)) '&'
('not' Pj(a,x) 'or' ('not' Pj(b,x) 'or' Pj(b,x))) by BINARITH:20
.=(TRUE 'or' 'not' Pj(b,x)) '&'
('not' Pj(a,x) 'or' ('not' Pj(b,x) 'or' Pj(b,x))) by BINARITH:18
.=(TRUE 'or' 'not' Pj(b,x)) '&'
('not' Pj(a,x) 'or' TRUE) by BINARITH:18
.=TRUE '&'
('not' Pj(a,x) 'or' TRUE) by BINARITH:19
.=TRUE '&' TRUE by BINARITH:19
.=TRUE by MARGREL1:45;
hence thesis;
end;
hence thesis by BVFUNC_1:def 14;
end;

theorem for a,b being Element of Funcs(Y,BOOLEAN) holds
(a '&' b) 'imp' a=I_el(Y)
proof
let a,b be Element of Funcs(Y,BOOLEAN);
for x being Element of Y holds Pj((a '&' b) 'imp' a,x)=TRUE
proof
let x be Element of Y;
Pj((a '&' b) 'imp' a,x)
='not' Pj(a '&' b,x) 'or' Pj(a,x) by BVFUNC_1:def 11
.='not'( Pj(a,x) '&' Pj(b,x)) 'or' Pj(a,x) by VALUAT_1:def 6
.=('not' Pj(a,x) 'or' 'not' Pj(b,x)) 'or' Pj(a,x) by BINARITH:9
.=('not' Pj(a,x) 'or' Pj(a,x)) 'or' 'not' Pj(b,x) by BINARITH:20
.=TRUE 'or' 'not' Pj(b,x) by BINARITH:18
.=TRUE by BINARITH:19;
hence thesis;
end;
hence thesis by BVFUNC_1:def 14;
end;

theorem for a,b being Element of Funcs(Y,BOOLEAN) holds
(a '&' b) 'imp' (a 'or' b)=I_el(Y)
proof
let a,b be Element of Funcs(Y,BOOLEAN);
for x being Element of Y holds Pj((a '&' b) 'imp' (a 'or' b),x)=TRUE
proof
let x be Element of Y;
Pj((a '&' b) 'imp' (a 'or' b),x)
='not' Pj(a '&' b,x) 'or' Pj(a 'or' b,x) by BVFUNC_1:def 11
.='not'( Pj(a,x) '&' Pj(b,x)) 'or' Pj(a 'or' b,x) by VALUAT_1:def 6
.=('not' Pj(a,x) 'or' 'not' Pj(b,x)) 'or' Pj(a 'or' b,x) by BINARITH:9
.=('not' Pj(a,x) 'or' 'not'
Pj(b,x)) 'or' (Pj(a,x) 'or' Pj(b,x)) by BVFUNC_1:def 7
.=(('not' Pj(b,x) 'or' 'not'
Pj(a,x)) 'or' Pj(a,x)) 'or' Pj(b,x) by BINARITH:20
.=('not' Pj(b,x) 'or' ('not'
Pj(a,x) 'or' Pj(a,x))) 'or' Pj(b,x) by BINARITH:20
.=('not' Pj(b,x) 'or' TRUE) 'or' Pj(b,x) by BINARITH:18
.=TRUE 'or' Pj(b,x) by BINARITH:19
.=TRUE by BINARITH:19;
hence thesis;
end;
hence thesis by BVFUNC_1:def 14;
end;

theorem for a,b being Element of Funcs(Y,BOOLEAN) holds
(a '&' b) 'imp' b=I_el(Y)
proof
let a,b be Element of Funcs(Y,BOOLEAN);
for x being Element of Y holds Pj((a '&' b) 'imp' b,x)=TRUE
proof
let x be Element of Y;
Pj((a '&' b) 'imp' b,x)
='not' Pj(a '&' b,x) 'or' Pj(b,x) by BVFUNC_1:def 11
.='not'( Pj(a,x) '&' Pj(b,x)) 'or' Pj(b,x) by VALUAT_1:def 6
.=('not' Pj(a,x) 'or' 'not' Pj(b,x)) 'or' Pj(b,x) by BINARITH:9
.='not' Pj(a,x) 'or' ('not' Pj(b,x) 'or' Pj(b,x)) by BINARITH:20
.='not' Pj(a,x) 'or' TRUE by BINARITH:18
.=TRUE by BINARITH:19;
hence thesis;
end;
hence thesis by BVFUNC_1:def 14;
end;

theorem for a being Element of Funcs(Y,BOOLEAN) holds
a 'imp' a '&' a=I_el(Y)
proof
let a be Element of Funcs(Y,BOOLEAN);
for x being Element of Y holds Pj(a 'imp' (a '&' a),x)=TRUE
proof
let x be Element of Y;
Pj(a 'imp' a '&' a,x)
='not' Pj(a,x) 'or' Pj(a '&' a,x) by BVFUNC_1:def 11
.='not' Pj(a,x) 'or' (Pj(a,x) '&' Pj(a,x)) by VALUAT_1:def 6
.=('not' Pj(a,x) 'or' Pj(a,x)) '&'
('not' Pj(a,x) 'or' Pj(a,x)) by BINARITH:23
.=TRUE '&'
('not' Pj(a,x) 'or' Pj(a,x)) by BINARITH:18
.=TRUE '&'
TRUE by BINARITH:18
.=TRUE by MARGREL1:45;
hence thesis;
end;
hence thesis by BVFUNC_1:def 14;
end;

theorem for a,b being Element of Funcs(Y,BOOLEAN) holds
(a 'eqv' b) 'imp' (a 'imp' b)=I_el(Y)
proof
let a,b be Element of Funcs(Y,BOOLEAN);
for x being Element of Y holds Pj((a 'eqv' b) 'imp' (a 'imp' b),x)=TRUE
proof
let x be Element of Y;
Pj((a 'eqv' b) 'imp' (a 'imp' b),x)
='not' Pj(a 'eqv' b,x) 'or' Pj(a 'imp' b,x) by BVFUNC_1:def 11
.='not' 'not'( Pj(a,x) 'xor' Pj(b,x)) 'or'
Pj(a 'imp' b,x) by BVFUNC_1:def 12
.=(Pj(a,x) 'xor' Pj(b,x)) 'or'
Pj(a 'imp' b,x) by MARGREL1:40
.=(('not' Pj(a,x) '&' Pj(b,x)) 'or' (Pj(a,x) '&' 'not' Pj(b,x))) 'or'
Pj(a 'imp' b,x) by BINARITH:def 2
.=(('not' Pj(a,x) '&' Pj(b,x)) 'or' (Pj(a,x) '&' 'not' Pj(b,x))) 'or'
('not' Pj(a,x) 'or' Pj(b,x)) by BVFUNC_1:def 11
.=(('not' Pj(a,x) '&' Pj(b,x)) 'or' (Pj(a,x) '&' 'not' Pj(b,x))) 'or'
('not' Pj(a,x) 'or' 'not' 'not' Pj(b,x)) by MARGREL1:40
.=(('not' Pj(a,x) '&' Pj(b,x)) 'or' (Pj(a,x) '&' 'not' Pj(b,x))) 'or'
'not'( Pj(a,x) '&' 'not' Pj(b,x)) by BINARITH:9
.=('not' Pj(a,x) '&' Pj(b,x)) 'or' ((Pj(a,x) '&' 'not' Pj(b,x)) 'or'
'not'( Pj(a,x) '&' 'not' Pj(b,x))) by BINARITH:20
.=('not' Pj(a,x) '&' Pj(b,x)) 'or' TRUE by BINARITH:18
.=TRUE by BINARITH:19;
hence thesis;
end;
hence thesis by BVFUNC_1:def 14;
end;

theorem for a,b being Element of Funcs(Y,BOOLEAN) holds
(a 'eqv' b) 'imp' (b 'imp' a)=I_el(Y)
proof
let a,b be Element of Funcs(Y,BOOLEAN);
for x being Element of Y holds Pj((a 'eqv' b) 'imp' (b 'imp' a),x)=TRUE
proof
let x be Element of Y;
Pj((a 'eqv' b) 'imp' (b 'imp' a),x)
='not' Pj(a 'eqv' b,x) 'or' Pj(b 'imp' a,x) by BVFUNC_1:def 11
.='not' 'not'( Pj(a,x) 'xor' Pj(b,x)) 'or'
Pj(b 'imp' a,x) by BVFUNC_1:def 12
.=(Pj(a,x) 'xor' Pj(b,x)) 'or'
Pj(b 'imp' a,x) by MARGREL1:40
.=(('not' Pj(a,x) '&' Pj(b,x)) 'or' (Pj(a,x) '&' 'not' Pj(b,x))) 'or'
Pj(b 'imp' a,x) by BINARITH:def 2
.=(('not' Pj(a,x) '&' Pj(b,x)) 'or' (Pj(a,x) '&' 'not' Pj(b,x))) 'or'
(Pj(a,x) 'or' 'not' Pj(b,x)) by BVFUNC_1:def 11
.=('not' Pj(a,x) '&' Pj(b,x)) 'or' ((Pj(a,x) '&' 'not' Pj(b,x)) 'or'
(Pj(a,x) 'or' 'not' Pj(b,x))) by BINARITH:20
.=('not' Pj(a,x) '&' Pj(b,x)) 'or' ((Pj(a,x) '&' 'not' Pj(b,x)) 'or'
('not' 'not' Pj(a,x) 'or' 'not' Pj(b,x))) by MARGREL1:40
.=('not' Pj(a,x) '&' Pj(b,x)) 'or' ('not'( 'not' Pj(a,x) '&' Pj(b,x)) 'or'
(Pj(a,x) '&' 'not' Pj(b,x))) by BINARITH:9
.=(('not' Pj(a,x) '&' Pj(b,x)) 'or' 'not'( 'not' Pj(a,x) '&' Pj(b,x))) 'or'
(Pj(a,x) '&' 'not' Pj(b,x)) by BINARITH:20
.=TRUE 'or' (Pj(a,x) '&' 'not' Pj(b,x)) by BINARITH:18
.=TRUE by BINARITH:19;
hence thesis;
end;
hence thesis by BVFUNC_1:def 14;
end;

theorem for a,b,c being Element of Funcs(Y,BOOLEAN) holds
((a 'or' b) 'or' c) 'imp' (a 'or' (b 'or' c))=I_el(Y)
proof
let a,b,c be Element of Funcs(Y,BOOLEAN);
for x being Element of Y holds
Pj(((a 'or' b) 'or' c) 'imp' (a 'or' (b 'or' c)),x)=TRUE
proof
let x be Element of Y;
Pj(((a 'or' b) 'or' c) 'imp' (a 'or' (b 'or' c)),x)
='not' Pj((a 'or' b) 'or' c,x) 'or' Pj(a 'or' (b 'or' c),x)
by BVFUNC_1:def 11
.='not'( Pj(a 'or' b,x) 'or' Pj(c,x)) 'or' Pj(a 'or' (b 'or' c),x)
by BVFUNC_1:def 7
.='not'( (Pj(a,x) 'or' Pj(b,x)) 'or' Pj(c,x)) 'or' Pj(a 'or' (b 'or' c),x)
by BVFUNC_1:def 7
.='not'( (Pj(a,x) 'or' Pj(b,x)) 'or' Pj(c,x)) 'or'
(Pj(a,x) 'or' Pj(b 'or' c,x))
by BVFUNC_1:def 7
.='not'( (Pj(a,x) 'or' Pj(b,x)) 'or' Pj(c,x)) 'or'
(Pj(a,x) 'or' (Pj(b,x) 'or' Pj(c,x)))
by BVFUNC_1:def 7
.='not'( (Pj(a,x) 'or' Pj(b,x)) 'or' Pj(c,x)) 'or'
((Pj(a,x) 'or' Pj(b,x)) 'or' Pj(c,x))
by BINARITH:20
.=TRUE by BINARITH:18;
hence thesis;
end;
hence thesis by BVFUNC_1:def 14;
end;

theorem for a,b,c being Element of Funcs(Y,BOOLEAN) holds
((a '&' b) '&' c) 'imp' (a '&' (b '&' c))=I_el(Y)
proof
let a,b,c be Element of Funcs(Y,BOOLEAN);
for x being Element of Y holds
Pj(((a '&' b) '&' c) 'imp' (a '&' (b '&' c)),x)=TRUE
proof
let x be Element of Y;
Pj(((a '&' b) '&' c) 'imp' (a '&' (b '&' c)),x)
='not' Pj((a '&' b) '&' c,x) 'or' Pj(a '&' (b '&' c),x)
by BVFUNC_1:def 11
.='not'( Pj(a '&' b,x) '&' Pj(c,x)) 'or' Pj(a '&' (b '&' c),x)
by VALUAT_1:def 6
.='not'( (Pj(a,x) '&' Pj(b,x)) '&' Pj(c,x)) 'or' Pj(a '&' (b '&' c),x)
by VALUAT_1:def 6
.='not'( (Pj(a,x) '&' Pj(b,x)) '&' Pj(c,x)) 'or'
(Pj(a,x) '&' Pj(b '&' c,x))
by VALUAT_1:def 6
.='not'( (Pj(a,x) '&' Pj(b,x)) '&' Pj(c,x)) 'or'
(Pj(a,x) '&' (Pj(b,x) '&' Pj(c,x)))
by VALUAT_1:def 6
.='not'( (Pj(a,x) '&' Pj(b,x)) '&' Pj(c,x)) 'or'
((Pj(a,x) '&' Pj(b,x)) '&' Pj(c,x))
by MARGREL1:52
.=TRUE by BINARITH:18;
hence thesis;
end;
hence thesis by BVFUNC_1:def 14;
end;

theorem for a,b,c being Element of Funcs(Y,BOOLEAN) holds
(a 'or' (b 'or' c)) 'imp' ((a 'or' b) 'or' c)=I_el(Y)
proof
let a,b,c be Element of Funcs(Y,BOOLEAN);
for x being Element of Y holds
Pj((a 'or' (b 'or' c)) 'imp' ((a 'or' b) 'or' c),x)=TRUE
proof
let x be Element of Y;
Pj((a 'or' (b 'or' c)) 'imp' ((a 'or' b) 'or' c),x)
='not' Pj(a 'or' (b 'or' c),x) 'or' Pj((a 'or' b) 'or' c,x)
by BVFUNC_1:def 11
.='not'( Pj(a,x) 'or' Pj(b 'or' c,x)) 'or' Pj((a 'or' b) 'or' c,x)
by BVFUNC_1:def 7
.='not'( Pj(a,x) 'or' (Pj(b,x) 'or' Pj(c,x))) 'or' Pj((a 'or' b) 'or' c,x)
by BVFUNC_1:def 7
.='not'( Pj(a,x) 'or' (Pj(b,x) 'or' Pj(c,x))) 'or'
(Pj(a 'or' b,x) 'or' Pj(c,x))
by BVFUNC_1:def 7
.='not'( Pj(a,x) 'or' (Pj(b,x) 'or' Pj(c,x))) 'or'
((Pj(a,x) 'or' Pj(b,x)) 'or' Pj(c,x))
by BVFUNC_1:def 7
.='not'( Pj(a,x) 'or' (Pj(b,x) 'or' Pj(c,x))) 'or'
(Pj(a,x) 'or' (Pj(b,x) 'or' Pj(c,x)))
by BINARITH:20
.=TRUE by BINARITH:18;
hence thesis;
end;
hence thesis by BVFUNC_1:def 14;
end;

```