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 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 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 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 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 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 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 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