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, y being Element of Union X st L is vf-qc-correct & L is subst-correct holds
(\ex (x,y,A)) \imp (\ex (y,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, y being Element of Union X st L is vf-qc-correct & L is subst-correct holds
(\ex (x,y,A)) \imp (\ex (y,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, y being Element of Union X st L is vf-qc-correct & L is subst-correct holds
(\ex (x,y,A)) \imp (\ex (y,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, y being Element of Union X st L is vf-qc-correct & L is subst-correct holds
(\ex (x,y,A)) \imp (\ex (y,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, y being Element of Union X st L is vf-qc-correct & L is subst-correct holds
(\ex (x,y,A)) \imp (\ex (y,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, y being Element of Union X st L is vf-qc-correct & L is subst-correct holds
(\ex (x,y,A)) \imp (\ex (y,x,A)) in G
let G be QC-theory of L; :: thesis: for A being Formula of L
for x, y being Element of Union X st L is vf-qc-correct & L is subst-correct holds
(\ex (x,y,A)) \imp (\ex (y,x,A)) in G
let A be Formula of L; :: thesis: for x, y being Element of Union X st L is vf-qc-correct & L is subst-correct holds
(\ex (x,y,A)) \imp (\ex (y,x,A)) in G
let x, y be Element of Union X; :: thesis: ( L is vf-qc-correct & L is subst-correct implies (\ex (x,y,A)) \imp (\ex (y,x,A)) in G )
assume A1: ( L is vf-qc-correct & L is subst-correct ) ; :: thesis: (\ex (x,y,A)) \imp (\ex (y,x,A)) in G
then (\for (y,x,(\not A))) \imp (\for (x,y,(\not A))) in G by Th138;
then A2: (\not (\for (x,y,(\not A)))) \imp (\not (\for (y,x,(\not A)))) in G by Th58;
( (\ex (x,y,A)) \iff (\not (\for (x,y,(\not A)))) in G & (\ex (y,x,A)) \iff (\not (\for (y,x,(\not A)))) in G ) by A1, Th111;
then ( (\ex (x,y,A)) \imp (\not (\for (y,x,(\not A)))) in G & (\not (\for (y,x,(\not A)))) \iff (\ex (y,x,A)) in G ) by A2, Th90, Th92;
hence (\ex (x,y,A)) \imp (\ex (y,x,A)) in G by Th93; :: thesis: verum