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
for a being SortSymbol of J st x in X . a & x nin (vf A) . a & \for (x,(A \imp B)) in G holds
A \imp (\for (x,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
for a being SortSymbol of J st x in X . a & x nin (vf A) . a & \for (x,(A \imp B)) in G holds
A \imp (\for (x,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
for a being SortSymbol of J st x in X . a & x nin (vf A) . a & \for (x,(A \imp B)) in G holds
A \imp (\for (x,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
for a being SortSymbol of J st x in X . a & x nin (vf A) . a & \for (x,(A \imp B)) in G holds
A \imp (\for (x,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
for a being SortSymbol of J st x in X . a & x nin (vf A) . a & \for (x,(A \imp B)) in G holds
A \imp (\for (x,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
for a being SortSymbol of J st x in X . a & x nin (vf A) . a & \for (x,(A \imp B)) in G holds
A \imp (\for (x,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
for a being SortSymbol of J st x in X . a & x nin (vf A) . a & \for (x,(A \imp B)) in G holds
A \imp (\for (x,B)) in G
let A, B be Formula of L; :: thesis: for x being Element of Union X
for a being SortSymbol of J st x in X . a & x nin (vf A) . a & \for (x,(A \imp B)) in G holds
A \imp (\for (x,B)) in G
let x be Element of Union X; :: thesis: for a being SortSymbol of J st x in X . a & x nin (vf A) . a & \for (x,(A \imp B)) in G holds
A \imp (\for (x,B)) in G
let a be SortSymbol of J; :: thesis: ( x in X . a & x nin (vf A) . a & \for (x,(A \imp B)) in G implies A \imp (\for (x,B)) in G )
assume A1: ( x in X . a & x nin (vf A) . a & \for (x,(A \imp B)) in G ) ; :: thesis: A \imp (\for (x,B)) in G
then (\for (x,(A \imp B))) \imp (A \imp (\for (x,B))) in G by Def39;
hence A \imp (\for (x,B)) in G by Def38, A1; :: thesis: verum