let n be non empty Nat; for J being non empty non void Signature
for T being non-empty MSAlgebra over J
for X being V3() GeneratorSet of T
for S1 being non empty non void b1 -extension n PC-correct QC-correct QCLangSignature over Union X
for L being non-empty Language of X extended_by ({}, the carrier of S1),S1
for G being QC-theory of L
for A being Formula of L
for x, y being Element of Union X st L is vf-qc-correct & L is subst-correct holds
(\for (x,y,A)) \imp (\for (y,x,A)) in G
let J be non empty non void Signature; for T being non-empty MSAlgebra over J
for X being V3() GeneratorSet of T
for S1 being non empty non void J -extension n PC-correct QC-correct QCLangSignature over Union X
for L being non-empty Language of X extended_by ({}, the carrier of S1),S1
for G being QC-theory of L
for A being Formula of L
for x, y being Element of Union X st L is vf-qc-correct & L is subst-correct holds
(\for (x,y,A)) \imp (\for (y,x,A)) in G
let T be non-empty MSAlgebra over J; for X being V3() GeneratorSet of T
for S1 being non empty non void J -extension n PC-correct QC-correct QCLangSignature over Union X
for L being non-empty Language of X extended_by ({}, the carrier of S1),S1
for G being QC-theory of L
for A being Formula of L
for x, y being Element of Union X st L is vf-qc-correct & L is subst-correct holds
(\for (x,y,A)) \imp (\for (y,x,A)) in G
let X be V3() GeneratorSet of T; for S1 being non empty non void J -extension n PC-correct QC-correct QCLangSignature over Union X
for L being non-empty Language of X extended_by ({}, the carrier of S1),S1
for G being QC-theory of L
for A being Formula of L
for x, y being Element of Union X st L is vf-qc-correct & L is subst-correct holds
(\for (x,y,A)) \imp (\for (y,x,A)) in G
let S1 be non empty non void J -extension n PC-correct QC-correct QCLangSignature over Union X; for L being non-empty Language of X extended_by ({}, the carrier of S1),S1
for G being QC-theory of L
for A being Formula of L
for x, y being Element of Union X st L is vf-qc-correct & L is subst-correct holds
(\for (x,y,A)) \imp (\for (y,x,A)) in G
let L be non-empty Language of X extended_by ({}, the carrier of S1),S1; for G being QC-theory of L
for A being Formula of L
for x, y being Element of Union X st L is vf-qc-correct & L is subst-correct holds
(\for (x,y,A)) \imp (\for (y,x,A)) in G
let G be QC-theory of L; for A being Formula of L
for x, y being Element of Union X st L is vf-qc-correct & L is subst-correct holds
(\for (x,y,A)) \imp (\for (y,x,A)) in G
let A be Formula of L; for x, y being Element of Union X st L is vf-qc-correct & L is subst-correct holds
(\for (x,y,A)) \imp (\for (y,x,A)) in G
let x, y be Element of Union X; ( L is vf-qc-correct & L is subst-correct implies (\for (x,y,A)) \imp (\for (y,x,A)) in G )
assume A1:
( L is vf-qc-correct & L is subst-correct )
; (\for (x,y,A)) \imp (\for (y,x,A)) in G
then
(\for (y,A)) \imp A in G
by Th104;
then A2:
\for (x,((\for (y,A)) \imp A)) in G
by Def39;
(\for (x,((\for (y,A)) \imp A))) \imp ((\for (x,(\for (y,A)))) \imp (\for (x,A))) in G
by A1, Th109;
then
(\for (x,(\for (y,A)))) \imp (\for (x,A)) in G
by A2, Def38;
then A3:
\for (y,((\for (x,(\for (y,A)))) \imp (\for (x,A)))) in G
by Def39;
consider a being object such that
A4:
( a in dom X & y in X . a )
by CARD_5:2;
consider b being object such that
A5:
( b in dom X & x in X . b )
by CARD_5:2;
J is Subsignature of S1
by Def2;
then
( dom X = the carrier of J & the carrier of J c= the carrier of S1 )
by PARTFUN1:def 2, INSTALG1:10;
then reconsider a = a, b = b as Element of S1 by A4, A5;
reconsider c = a as Element of J by A4;
vf (\for (x,y,A)) = (vf (\for (y,A))) (\) (b -singleton x)
by A1, A5;
then
(vf (\for (x,y,A))) . a = ((vf (\for (y,A))) . a) \ ((b -singleton x) . a)
by PBOOLE:def 6;
then
y nin (vf (\for (x,y,A))) . c
by A1, A4, Th113;
hence
(\for (x,y,A)) \imp (\for (y,x,A)) in G
by A3, A4, Th108; verum