let Al be QC-alphabet ; :: thesis: for p being Element of CQC-WFF Al

for x being bound_QC-variable of Al

for f being FinSequence of CQC-WFF Al st Suc f = All (x,p) & Ant f |= Suc f holds

for y being bound_QC-variable of Al holds Ant f |= p . (x,y)

let p be Element of CQC-WFF Al; :: thesis: for x being bound_QC-variable of Al

for f being FinSequence of CQC-WFF Al st Suc f = All (x,p) & Ant f |= Suc f holds

for y being bound_QC-variable of Al holds Ant f |= p . (x,y)

let x be bound_QC-variable of Al; :: thesis: for f being FinSequence of CQC-WFF Al st Suc f = All (x,p) & Ant f |= Suc f holds

for y being bound_QC-variable of Al holds Ant f |= p . (x,y)

let f be FinSequence of CQC-WFF Al; :: thesis: ( Suc f = All (x,p) & Ant f |= Suc f implies for y being bound_QC-variable of Al holds Ant f |= p . (x,y) )

assume A1: ( Suc f = All (x,p) & Ant f |= Suc f ) ; :: thesis: for y being bound_QC-variable of Al holds Ant f |= p . (x,y)

let y be bound_QC-variable of Al; :: thesis: Ant f |= p . (x,y)

let A be non empty set ; :: according to CALCUL_1:def 15 :: thesis: for J being interpretation of Al,A

for v being Element of Valuations_in (Al,A) st J,v |= Ant f holds

J,v |= p . (x,y)

let J be interpretation of Al,A; :: thesis: for v being Element of Valuations_in (Al,A) st J,v |= Ant f holds

J,v |= p . (x,y)

let v be Element of Valuations_in (Al,A); :: thesis: ( J,v |= Ant f implies J,v |= p . (x,y) )

assume J,v |= Ant f ; :: thesis: J,v |= p . (x,y)

then A2: J,v |= All (x,p) by A1;

ex a being Element of A st

( v . y = a & J,v . (x | a) |= p )

for x being bound_QC-variable of Al

for f being FinSequence of CQC-WFF Al st Suc f = All (x,p) & Ant f |= Suc f holds

for y being bound_QC-variable of Al holds Ant f |= p . (x,y)

let p be Element of CQC-WFF Al; :: thesis: for x being bound_QC-variable of Al

for f being FinSequence of CQC-WFF Al st Suc f = All (x,p) & Ant f |= Suc f holds

for y being bound_QC-variable of Al holds Ant f |= p . (x,y)

let x be bound_QC-variable of Al; :: thesis: for f being FinSequence of CQC-WFF Al st Suc f = All (x,p) & Ant f |= Suc f holds

for y being bound_QC-variable of Al holds Ant f |= p . (x,y)

let f be FinSequence of CQC-WFF Al; :: thesis: ( Suc f = All (x,p) & Ant f |= Suc f implies for y being bound_QC-variable of Al holds Ant f |= p . (x,y) )

assume A1: ( Suc f = All (x,p) & Ant f |= Suc f ) ; :: thesis: for y being bound_QC-variable of Al holds Ant f |= p . (x,y)

let y be bound_QC-variable of Al; :: thesis: Ant f |= p . (x,y)

let A be non empty set ; :: according to CALCUL_1:def 15 :: thesis: for J being interpretation of Al,A

for v being Element of Valuations_in (Al,A) st J,v |= Ant f holds

J,v |= p . (x,y)

let J be interpretation of Al,A; :: thesis: for v being Element of Valuations_in (Al,A) st J,v |= Ant f holds

J,v |= p . (x,y)

let v be Element of Valuations_in (Al,A); :: thesis: ( J,v |= Ant f implies J,v |= p . (x,y) )

assume J,v |= Ant f ; :: thesis: J,v |= p . (x,y)

then A2: J,v |= All (x,p) by A1;

ex a being Element of A st

( v . y = a & J,v . (x | a) |= p )

proof

hence
J,v |= p . (x,y)
by Th24; :: thesis: verum
take
v . y
; :: thesis: ( v . y = v . y & J,v . (x | (v . y)) |= p )

thus ( v . y = v . y & J,v . (x | (v . y)) |= p ) by A2, SUBLEMMA:50; :: thesis: verum

end;thus ( v . y = v . y & J,v . (x | (v . y)) |= p ) by A2, SUBLEMMA:50; :: thesis: verum