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 holds (\for (x,A)) \iff (\not (\ex (x,(\not 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 holds (\for (x,A)) \iff (\not (\ex (x,(\not 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 holds (\for (x,A)) \iff (\not (\ex (x,(\not 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 holds (\for (x,A)) \iff (\not (\ex (x,(\not 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 holds (\for (x,A)) \iff (\not (\ex (x,(\not 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 holds (\for (x,A)) \iff (\not (\ex (x,(\not A)))) in G
let G be QC-theory of L; :: thesis: for A being Formula of L
for x being Element of Union X holds (\for (x,A)) \iff (\not (\ex (x,(\not A)))) in G
let A be Formula of L; :: thesis: for x being Element of Union X holds (\for (x,A)) \iff (\not (\ex (x,(\not A)))) in G
let x be Element of Union X; :: thesis: (\for (x,A)) \iff (\not (\ex (x,(\not A)))) in G
(\ex (x,(\not A))) \iff (\not (\for (x,A))) in G by Def39;
then (\not (\for (x,A))) \iff (\ex (x,(\not A))) in G by Th90;
then (\not (\not (\for (x,A)))) \iff (\not (\ex (x,(\not A)))) in G by Th94;
hence (\for (x,A)) \iff (\not (\ex (x,(\not A)))) in G by Th95; :: thesis: verum