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, y being Element of Union X st L is subst-correct & L is vf-qc-correct holds

(\for (x,y,A)) \iff (\not (\ex (x,y,(\not 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, y being Element of Union X st L is subst-correct & L is vf-qc-correct holds

(\for (x,y,A)) \iff (\not (\ex (x,y,(\not 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, y being Element of Union X st L is subst-correct & L is vf-qc-correct holds

(\for (x,y,A)) \iff (\not (\ex (x,y,(\not 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, y being Element of Union X st L is subst-correct & L is vf-qc-correct holds

(\for (x,y,A)) \iff (\not (\ex (x,y,(\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, y being Element of Union X st L is subst-correct & L is vf-qc-correct holds

(\for (x,y,A)) \iff (\not (\ex (x,y,(\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, y being Element of Union X st L is subst-correct & L is vf-qc-correct holds

(\for (x,y,A)) \iff (\not (\ex (x,y,(\not 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 subst-correct & L is vf-qc-correct holds

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

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

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

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

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

A2: (\for (x,y,A)) \iff (\not (\ex (x,(\not (\for (y,A)))))) in G by Th106;

( (\for (y,A)) \iff (\not (\ex (y,(\not A)))) in G & (\ex (y,(\not A))) \iff (\not (\not (\ex (y,(\not A))))) in G ) by Th66, Th106;

then ( (\not (\for (y,A))) \iff (\not (\not (\ex (y,(\not A))))) in G & (\not (\not (\ex (y,(\not A))))) \iff (\ex (y,(\not A))) in G ) by Th90, Th94;

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

then ( (\not (\for (y,A))) \imp (\ex (y,(\not A))) in G & (\ex (y,(\not A))) \imp (\not (\for (y,A))) in G ) by Th43;

then A3: ( \for (x,((\not (\for (y,A))) \imp (\ex (y,(\not A))))) in G & \for (x,((\ex (y,(\not A))) \imp (\not (\for (y,A))))) in G ) by Def39;

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

then ( (\ex (x,(\not (\for (y,A))))) \imp (\ex (x,(\ex (y,(\not A))))) in G & (\ex (x,(\ex (y,(\not A))))) \imp (\ex (x,(\not (\for (y,A))))) in G ) by A3, Def38;

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

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

hence (\for (x,y,A)) \iff (\not (\ex (x,y,(\not A)))) in G by A2, Th91; :: 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, y being Element of Union X st L is subst-correct & L is vf-qc-correct holds

(\for (x,y,A)) \iff (\not (\ex (x,y,(\not 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, y being Element of Union X st L is subst-correct & L is vf-qc-correct holds

(\for (x,y,A)) \iff (\not (\ex (x,y,(\not 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, y being Element of Union X st L is subst-correct & L is vf-qc-correct holds

(\for (x,y,A)) \iff (\not (\ex (x,y,(\not 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, y being Element of Union X st L is subst-correct & L is vf-qc-correct holds

(\for (x,y,A)) \iff (\not (\ex (x,y,(\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, y being Element of Union X st L is subst-correct & L is vf-qc-correct holds

(\for (x,y,A)) \iff (\not (\ex (x,y,(\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, y being Element of Union X st L is subst-correct & L is vf-qc-correct holds

(\for (x,y,A)) \iff (\not (\ex (x,y,(\not 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 subst-correct & L is vf-qc-correct holds

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

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

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

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

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

A2: (\for (x,y,A)) \iff (\not (\ex (x,(\not (\for (y,A)))))) in G by Th106;

( (\for (y,A)) \iff (\not (\ex (y,(\not A)))) in G & (\ex (y,(\not A))) \iff (\not (\not (\ex (y,(\not A))))) in G ) by Th66, Th106;

then ( (\not (\for (y,A))) \iff (\not (\not (\ex (y,(\not A))))) in G & (\not (\not (\ex (y,(\not A))))) \iff (\ex (y,(\not A))) in G ) by Th90, Th94;

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

then ( (\not (\for (y,A))) \imp (\ex (y,(\not A))) in G & (\ex (y,(\not A))) \imp (\not (\for (y,A))) in G ) by Th43;

then A3: ( \for (x,((\not (\for (y,A))) \imp (\ex (y,(\not A))))) in G & \for (x,((\ex (y,(\not A))) \imp (\not (\for (y,A))))) in G ) by Def39;

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

then ( (\ex (x,(\not (\for (y,A))))) \imp (\ex (x,(\ex (y,(\not A))))) in G & (\ex (x,(\ex (y,(\not A))))) \imp (\ex (x,(\not (\for (y,A))))) in G ) by A3, Def38;

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

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

hence (\for (x,y,A)) \iff (\not (\ex (x,y,(\not A)))) in G by A2, Th91; :: thesis: verum