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, B being Formula of L
for x being Element of Union X st L is subst-correct & L is vf-qc-correct holds
(\for (x,((\not A) \imp (\not B)))) \imp ((\for (x,B)) \imp (\for (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, B being Formula of L
for x being Element of Union X st L is subst-correct & L is vf-qc-correct holds
(\for (x,((\not A) \imp (\not B)))) \imp ((\for (x,B)) \imp (\for (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, B being Formula of L
for x being Element of Union X st L is subst-correct & L is vf-qc-correct holds
(\for (x,((\not A) \imp (\not B)))) \imp ((\for (x,B)) \imp (\for (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, B being Formula of L
for x being Element of Union X st L is subst-correct & L is vf-qc-correct holds
(\for (x,((\not A) \imp (\not B)))) \imp ((\for (x,B)) \imp (\for (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, B being Formula of L
for x being Element of Union X st L is subst-correct & L is vf-qc-correct holds
(\for (x,((\not A) \imp (\not B)))) \imp ((\for (x,B)) \imp (\for (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, B being Formula of L
for x being Element of Union X st L is subst-correct & L is vf-qc-correct holds
(\for (x,((\not A) \imp (\not B)))) \imp ((\for (x,B)) \imp (\for (x,A))) in G
let G be QC-theory of L; for A, B being Formula of L
for x being Element of Union X st L is subst-correct & L is vf-qc-correct holds
(\for (x,((\not A) \imp (\not B)))) \imp ((\for (x,B)) \imp (\for (x,A))) in G
let A, B be Formula of L; for x being Element of Union X st L is subst-correct & L is vf-qc-correct holds
(\for (x,((\not A) \imp (\not B)))) \imp ((\for (x,B)) \imp (\for (x,A))) in G
let x be Element of Union X; ( L is subst-correct & L is vf-qc-correct implies (\for (x,((\not A) \imp (\not B)))) \imp ((\for (x,B)) \imp (\for (x,A))) in G )
assume A1:
( L is subst-correct & L is vf-qc-correct )
; (\for (x,((\not A) \imp (\not B)))) \imp ((\for (x,B)) \imp (\for (x,A))) in G
((\not A) \imp (\not B)) \imp (B \imp A) in G
by Def38;
then A2:
(\for (x,((\not A) \imp (\not B)))) \imp (\for (x,(B \imp A))) in G
by A1, Th115;
(\for (x,(B \imp A))) \imp ((\for (x,B)) \imp (\for (x,A))) in G
by A1, Th109;
hence
(\for (x,((\not A) \imp (\not B)))) \imp ((\for (x,B)) \imp (\for (x,A))) in G
by A2, Th45; verum