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 being Formula of L

for x being Element of Union X st L is subst-correct & L is vf-qc-correct holds

(\for (x,(A \and A))) \iff (\for (x,A)) 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 being Formula of L

for x being Element of Union X st L is subst-correct & L is vf-qc-correct holds

(\for (x,(A \and A))) \iff (\for (x,A)) 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 being Formula of L

for x being Element of Union X st L is subst-correct & L is vf-qc-correct holds

(\for (x,(A \and A))) \iff (\for (x,A)) 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 being Formula of L

for x being Element of Union X st L is subst-correct & L is vf-qc-correct holds

(\for (x,(A \and A))) \iff (\for (x,A)) 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 being Formula of L

for x being Element of Union X st L is subst-correct & L is vf-qc-correct holds

(\for (x,(A \and A))) \iff (\for (x,A)) 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 being Formula of L

for x being Element of Union X st L is subst-correct & L is vf-qc-correct holds

(\for (x,(A \and A))) \iff (\for (x,A)) in G

let G be QC-theory of L; :: thesis: for A being Formula of L

for x being Element of Union X st L is subst-correct & L is vf-qc-correct holds

(\for (x,(A \and A))) \iff (\for (x,A)) in G

let A be Formula of L; :: thesis: for x being Element of Union X st L is subst-correct & L is vf-qc-correct holds

(\for (x,(A \and A))) \iff (\for (x,A)) in G

let x be Element of Union X; :: thesis: ( L is subst-correct & L is vf-qc-correct implies (\for (x,(A \and A))) \iff (\for (x,A)) in G )

assume A1: ( L is subst-correct & L is vf-qc-correct ) ; :: thesis: (\for (x,(A \and A))) \iff (\for (x,A)) in G

( (A \and A) \imp A in G & A \imp (A \and A) in G ) by Def38, Th53;

then A2: ( \for (x,((A \and A) \imp A)) in G & \for (x,(A \imp (A \and A))) in G ) by Def39;

( (\for (x,(A \imp (A \and A)))) \imp ((\for (x,A)) \imp (\for (x,(A \and A)))) in G & (\for (x,((A \and A) \imp A))) \imp ((\for (x,(A \and A))) \imp (\for (x,A))) in G ) by A1, Th109;

then ( (\for (x,A)) \imp (\for (x,(A \and A))) in G & (\for (x,(A \and A))) \imp (\for (x,A)) in G ) by A2, Def38;

hence (\for (x,(A \and A))) \iff (\for (x,A)) in G by Th43; :: 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 being Formula of L

for x being Element of Union X st L is subst-correct & L is vf-qc-correct holds

(\for (x,(A \and A))) \iff (\for (x,A)) 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 being Formula of L

for x being Element of Union X st L is subst-correct & L is vf-qc-correct holds

(\for (x,(A \and A))) \iff (\for (x,A)) 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 being Formula of L

for x being Element of Union X st L is subst-correct & L is vf-qc-correct holds

(\for (x,(A \and A))) \iff (\for (x,A)) 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 being Formula of L

for x being Element of Union X st L is subst-correct & L is vf-qc-correct holds

(\for (x,(A \and A))) \iff (\for (x,A)) 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 being Formula of L

for x being Element of Union X st L is subst-correct & L is vf-qc-correct holds

(\for (x,(A \and A))) \iff (\for (x,A)) 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 being Formula of L

for x being Element of Union X st L is subst-correct & L is vf-qc-correct holds

(\for (x,(A \and A))) \iff (\for (x,A)) in G

let G be QC-theory of L; :: thesis: for A being Formula of L

for x being Element of Union X st L is subst-correct & L is vf-qc-correct holds

(\for (x,(A \and A))) \iff (\for (x,A)) in G

let A be Formula of L; :: thesis: for x being Element of Union X st L is subst-correct & L is vf-qc-correct holds

(\for (x,(A \and A))) \iff (\for (x,A)) in G

let x be Element of Union X; :: thesis: ( L is subst-correct & L is vf-qc-correct implies (\for (x,(A \and A))) \iff (\for (x,A)) in G )

assume A1: ( L is subst-correct & L is vf-qc-correct ) ; :: thesis: (\for (x,(A \and A))) \iff (\for (x,A)) in G

( (A \and A) \imp A in G & A \imp (A \and A) in G ) by Def38, Th53;

then A2: ( \for (x,((A \and A) \imp A)) in G & \for (x,(A \imp (A \and A))) in G ) by Def39;

( (\for (x,(A \imp (A \and A)))) \imp ((\for (x,A)) \imp (\for (x,(A \and A)))) in G & (\for (x,((A \and A) \imp A))) \imp ((\for (x,(A \and A))) \imp (\for (x,A))) in G ) by A1, Th109;

then ( (\for (x,A)) \imp (\for (x,(A \and A))) in G & (\for (x,(A \and A))) \imp (\for (x,A)) in G ) by A2, Def38;

hence (\for (x,(A \and A))) \iff (\for (x,A)) in G by Th43; :: thesis: verum