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 V3() GeneratorSet of T

for S1 being non empty non void b_{1} -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 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 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 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 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 V3() 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

for T being non-empty MSAlgebra over J

for X being V3() GeneratorSet of T

for S1 being non empty non void b

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 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 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 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 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 V3() 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