let n be non empty Nat; :: thesis: for J being non empty non void Signature
for T being non-empty MSAlgebra over J
for X being empty-yielding 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 a being SortSymbol of J st x in X . a & x nin (vf B) . a holds
(\for (x,(A \imp B))) \imp ((\ex (x,A)) \imp B) in G
let J be non empty non void Signature; :: thesis: for T being non-empty MSAlgebra over J
for X being empty-yielding 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 a being SortSymbol of J st x in X . a & x nin (vf B) . a holds
(\for (x,(A \imp B))) \imp ((\ex (x,A)) \imp B) in G
let T be non-empty MSAlgebra over J; :: thesis: for X being empty-yielding 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 a being SortSymbol of J st x in X . a & x nin (vf B) . a holds
(\for (x,(A \imp B))) \imp ((\ex (x,A)) \imp B) in G
let X be empty-yielding GeneratorSet of T; :: thesis: 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 a being SortSymbol of J st x in X . a & x nin (vf B) . a holds
(\for (x,(A \imp B))) \imp ((\ex (x,A)) \imp B) in G
let S1 be non empty non void J -extension n PC-correct QC-correct QCLangSignature over Union X; :: thesis: 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 a being SortSymbol of J st x in X . a & x nin (vf B) . a holds
(\for (x,(A \imp B))) \imp ((\ex (x,A)) \imp B) in G
let L be non-empty Language of X extended_by ({}, the carrier of S1),S1; :: thesis: 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 a being SortSymbol of J st x in X . a & x nin (vf B) . a holds
(\for (x,(A \imp B))) \imp ((\ex (x,A)) \imp B) in G
let G be QC-theory of L; :: thesis: 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 a being SortSymbol of J st x in X . a & x nin (vf B) . a holds
(\for (x,(A \imp B))) \imp ((\ex (x,A)) \imp B) in G
let A, B be Formula of L; :: thesis: for x being Element of Union X st L is subst-correct & L is vf-qc-correct holds
for a being SortSymbol of J st x in X . a & x nin (vf B) . a holds
(\for (x,(A \imp B))) \imp ((\ex (x,A)) \imp B) in G
let x be Element of Union X; :: thesis: ( L is subst-correct & L is vf-qc-correct implies for a being SortSymbol of J st x in X . a & x nin (vf B) . a holds
(\for (x,(A \imp B))) \imp ((\ex (x,A)) \imp B) in G )
set Y = X extended_by ({}, the carrier of S1);
assume A1: ( L is subst-correct & L is vf-qc-correct ) ; :: thesis: for a being SortSymbol of J st x in X . a & x nin (vf B) . a holds
(\for (x,(A \imp B))) \imp ((\ex (x,A)) \imp B) in G
let a be SortSymbol of J; :: thesis: ( x in X . a & x nin (vf B) . a implies (\for (x,(A \imp B))) \imp ((\ex (x,A)) \imp B) in G )
assume A2: ( x in X . a & x nin (vf B) . a ) ; :: thesis: (\for (x,(A \imp B))) \imp ((\ex (x,A)) \imp B) in G
A3: (A \imp B) \imp ((\not B) \imp (\not A)) in G by Th57;
A4: (\for (x,(A \imp B))) \imp (\for (x,((\not B) \imp (\not A)))) in G by A1, A3, Th115;
x nin (vf (\not B)) . a by A1, A2;
then (\for (x,((\not B) \imp (\not A)))) \imp ((\not B) \imp (\for (x,(\not A)))) in G by A2, Def39;
then A5: (\for (x,(A \imp B))) \imp ((\not B) \imp (\for (x,(\not A)))) in G by A4, Th45;
( (\not (\ex (x,A))) \iff (\for (x,(\not A))) in G & ((\not (\ex (x,A))) \iff (\for (x,(\not A)))) \imp ((\for (x,(\not A))) \imp (\not (\ex (x,A)))) in G ) by Def39, Def38;
then (\for (x,(\not A))) \imp (\not (\ex (x,A))) in G by Def38;
then A6: (\for (x,(A \imp B))) \imp ((\not B) \imp (\not (\ex (x,A)))) in G by A5, Th69;
((\not B) \imp (\not (\ex (x,A)))) \imp ((\ex (x,A)) \imp B) in G by Def38;
hence (\for (x,(A \imp B))) \imp ((\ex (x,A)) \imp B) in G by A6, Th45; :: thesis: verum