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,(\not A))) \iff (\not (\ex (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,(\not A))) \iff (\not (\ex (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,(\not A))) \iff (\not (\ex (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,(\not A))) \iff (\not (\ex (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,(\not A))) \iff (\not (\ex (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,(\not A))) \iff (\not (\ex (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,(\not A))) \iff (\not (\ex (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,(\not A))) \iff (\not (\ex (x,A))) in G

let x be Element of Union X; :: thesis: ( L is subst-correct & L is vf-qc-correct implies (\for (x,(\not A))) \iff (\not (\ex (x,A))) in G )

assume A1: ( L is subst-correct & L is vf-qc-correct ) ; :: thesis: (\for (x,(\not A))) \iff (\not (\ex (x,A))) in G

A \imp (\not (\not A)) in G by Th64;

then ( \for (x,(A \imp (\not (\not A)))) in G & (\for (x,(A \imp (\not (\not A))))) \imp ((\ex (x,A)) \imp (\ex (x,(\not (\not A))))) in G ) by A1, Def39, Th121;

then A2: (\ex (x,A)) \imp (\ex (x,(\not (\not A)))) in G by Def38;

(\not (\not A)) \imp A in G by Th65;

then ( \for (x,((\not (\not A)) \imp A)) in G & (\for (x,((\not (\not A)) \imp A))) \imp ((\ex (x,(\not (\not A)))) \imp (\ex (x,A))) in G ) by A1, Def39, Th121;

then (\ex (x,(\not (\not A)))) \imp (\ex (x,A)) in G by Def38;

then ( (\ex (x,A)) \iff (\ex (x,(\not (\not A)))) in G & (\ex (x,(\not (\not A)))) \iff (\not (\for (x,(\not A)))) in G ) by Def39, A2, Th43;

then (\ex (x,A)) \iff (\not (\for (x,(\not A)))) in G by Th91;

then (\not (\for (x,(\not A)))) \iff (\ex (x,A)) in G by Th90;

then (\not (\not (\for (x,(\not A))))) \iff (\not (\ex (x,A))) in G by Th94;

hence (\for (x,(\not A))) \iff (\not (\ex (x,A))) in G by Th95; :: 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,(\not A))) \iff (\not (\ex (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,(\not A))) \iff (\not (\ex (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,(\not A))) \iff (\not (\ex (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,(\not A))) \iff (\not (\ex (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,(\not A))) \iff (\not (\ex (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,(\not A))) \iff (\not (\ex (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,(\not A))) \iff (\not (\ex (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,(\not A))) \iff (\not (\ex (x,A))) in G

let x be Element of Union X; :: thesis: ( L is subst-correct & L is vf-qc-correct implies (\for (x,(\not A))) \iff (\not (\ex (x,A))) in G )

assume A1: ( L is subst-correct & L is vf-qc-correct ) ; :: thesis: (\for (x,(\not A))) \iff (\not (\ex (x,A))) in G

A \imp (\not (\not A)) in G by Th64;

then ( \for (x,(A \imp (\not (\not A)))) in G & (\for (x,(A \imp (\not (\not A))))) \imp ((\ex (x,A)) \imp (\ex (x,(\not (\not A))))) in G ) by A1, Def39, Th121;

then A2: (\ex (x,A)) \imp (\ex (x,(\not (\not A)))) in G by Def38;

(\not (\not A)) \imp A in G by Th65;

then ( \for (x,((\not (\not A)) \imp A)) in G & (\for (x,((\not (\not A)) \imp A))) \imp ((\ex (x,(\not (\not A)))) \imp (\ex (x,A))) in G ) by A1, Def39, Th121;

then (\ex (x,(\not (\not A)))) \imp (\ex (x,A)) in G by Def38;

then ( (\ex (x,A)) \iff (\ex (x,(\not (\not A)))) in G & (\ex (x,(\not (\not A)))) \iff (\not (\for (x,(\not A)))) in G ) by Def39, A2, Th43;

then (\ex (x,A)) \iff (\not (\for (x,(\not A)))) in G by Th91;

then (\not (\for (x,(\not A)))) \iff (\ex (x,A)) in G by Th90;

then (\not (\not (\for (x,(\not A))))) \iff (\not (\ex (x,A))) in G by Th94;

hence (\for (x,(\not A))) \iff (\not (\ex (x,A))) in G by Th95; :: thesis: verum