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 L is subst-correct & L is vf-qc-correct & x in X . a & x nin (vf A) . a holds
(\ex (x,(A \and B))) \iff (A \and (\ex (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 L is subst-correct & L is vf-qc-correct & x in X . a & x nin (vf A) . a holds
(\ex (x,(A \and B))) \iff (A \and (\ex (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 L is subst-correct & L is vf-qc-correct & x in X . a & x nin (vf A) . a holds
(\ex (x,(A \and B))) \iff (A \and (\ex (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 L is subst-correct & L is vf-qc-correct & x in X . a & x nin (vf A) . a holds
(\ex (x,(A \and B))) \iff (A \and (\ex (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 L is subst-correct & L is vf-qc-correct & x in X . a & x nin (vf A) . a holds
(\ex (x,(A \and B))) \iff (A \and (\ex (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 L is subst-correct & L is vf-qc-correct & x in X . a & x nin (vf A) . a holds
(\ex (x,(A \and B))) \iff (A \and (\ex (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 L is subst-correct & L is vf-qc-correct & x in X . a & x nin (vf A) . a holds
(\ex (x,(A \and B))) \iff (A \and (\ex (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 L is subst-correct & L is vf-qc-correct & x in X . a & x nin (vf A) . a holds
(\ex (x,(A \and B))) \iff (A \and (\ex (x,B))) in G
let x be Element of Union X; :: thesis: for a being SortSymbol of J st L is subst-correct & L is vf-qc-correct & x in X . a & x nin (vf A) . a holds
(\ex (x,(A \and B))) \iff (A \and (\ex (x,B))) in G
let a be SortSymbol of J; :: thesis: ( L is subst-correct & L is vf-qc-correct & x in X . a & x nin (vf A) . a implies (\ex (x,(A \and B))) \iff (A \and (\ex (x,B))) in G )
set Y = X extended_by ({}, the carrier of S1);
assume A1: ( L is subst-correct & L is vf-qc-correct ) ; :: thesis: ( not x in X . a or not x nin (vf A) . a or (\ex (x,(A \and B))) \iff (A \and (\ex (x,B))) in G )
assume A2: x in X . a ; :: thesis: ( not x nin (vf A) . a or (\ex (x,(A \and B))) \iff (A \and (\ex (x,B))) in G )
assume A3: x nin (vf A) . a ; :: thesis: (\ex (x,(A \and B))) \iff (A \and (\ex (x,B))) in G
vf (\not A) = vf A by A1;
then (\for (x,((\not A) \or (\not B)))) \imp ((\not A) \or (\for (x,(\not B)))) in G by A1, A2, A3, Th133;
then A4: (\not ((\not A) \or (\for (x,(\not B))))) \imp (\not (\for (x,((\not A) \or (\not B))))) in G by Th58;
(\for (x,(\not B))) \iff (\not (\ex (x,B))) in G by A1, Th122;
then ( (\not A) \imp (\not A) in G & (\for (x,(\not B))) \imp (\not (\ex (x,B))) in G ) by Th34, Th43;
then ( ((\not A) \or (\not (\ex (x,B)))) \imp (\not (A \and (\ex (x,B)))) in G & ((\not A) \or (\for (x,(\not B)))) \imp ((\not A) \or (\not (\ex (x,B)))) in G & ((\not A) \or (\not (\ex (x,B)))) \imp (\not (A \and (\ex (x,B)))) in G ) by Th59, Th73;
then ( (A \and (\ex (x,B))) \imp (\not ((\not A) \or (\not (\ex (x,B))))) in G & (\not ((\not A) \or (\not (\ex (x,B))))) \imp (\not ((\not A) \or (\for (x,(\not B))))) in G ) by Th58, Th67;
then (A \and (\ex (x,B))) \imp (\not ((\not A) \or (\for (x,(\not B))))) in G by Th45;
then A5: (A \and (\ex (x,B))) \imp (\not (\for (x,((\not A) \or (\not B))))) in G by A4, Th45;
( (\ex (x,(A \and B))) \iff (\not (\for (x,(\not (A \and B))))) in G & (\not (A \and B)) \imp ((\not A) \or (\not B)) in G ) by Th70, Th105;
then ( (\not (\for (x,(\not (A \and B))))) \imp (\ex (x,(A \and B))) in G & \for (x,((\not (A \and B)) \imp ((\not A) \or (\not B)))) in G & (\for (x,((\not (A \and B)) \imp ((\not A) \or (\not B))))) \imp ((\for (x,(\not (A \and B)))) \imp (\for (x,((\not A) \or (\not B))))) in G ) by A1, Def39, Th43, Th109;
then ( (\not (\ex (x,(A \and B)))) \imp (\for (x,(\not (A \and B)))) in G & (\for (x,(\not (A \and B)))) \imp (\for (x,((\not A) \or (\not B)))) in G ) by Def38, Th68;
then (\not (\ex (x,(A \and B)))) \imp (\for (x,((\not A) \or (\not B)))) in G by Th45;
then (\not (\for (x,((\not A) \or (\not B))))) \imp (\ex (x,(A \and B))) in G by Th68;
then A6: (A \and (\ex (x,B))) \imp (\ex (x,(A \and B))) in G by A5, Th45;
(\ex (x,(A \and B))) \imp (A \and (\ex (x,B))) in G by Th134, A1, A2, A3;
hence (\ex (x,(A \and B))) \iff (A \and (\ex (x,B))) in G by A6, Th43; :: thesis: verum