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 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, B being Formula of L
for x being Element of Union X st L is subst-correct & L is vf-qc-correct holds
(\for (x,(A \imp B))) \imp ((\for (x,A)) \imp (\for (x,B))) 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, B being Formula of L
for x being Element of Union X st L is subst-correct & L is vf-qc-correct holds
(\for (x,(A \imp B))) \imp ((\for (x,A)) \imp (\for (x,B))) 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, B being Formula of L
for x being Element of Union X st L is subst-correct & L is vf-qc-correct holds
(\for (x,(A \imp B))) \imp ((\for (x,A)) \imp (\for (x,B))) 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, B being Formula of L
for x being Element of Union X st L is subst-correct & L is vf-qc-correct holds
(\for (x,(A \imp B))) \imp ((\for (x,A)) \imp (\for (x,B))) 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, B being Formula of L
for x being Element of Union X st L is subst-correct & L is vf-qc-correct holds
(\for (x,(A \imp B))) \imp ((\for (x,A)) \imp (\for (x,B))) 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, B being Formula of L
for x being Element of Union X st L is subst-correct & L is vf-qc-correct holds
(\for (x,(A \imp B))) \imp ((\for (x,A)) \imp (\for (x,B))) in G

let G be QC-theory of L; :: thesis: for A, B being Formula of L
for x being Element of Union X st L is subst-correct & L is vf-qc-correct holds
(\for (x,(A \imp B))) \imp ((\for (x,A)) \imp (\for (x,B))) in G

let A, B 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,(A \imp B))) \imp ((\for (x,A)) \imp (\for (x,B))) in G

let x be Element of Union X; :: thesis: ( L is subst-correct & L is vf-qc-correct implies (\for (x,(A \imp B))) \imp ((\for (x,A)) \imp (\for (x,B))) in G )
set Y = X extended_by ({}, the carrier of S1);
consider a being object such that
A1: ( a in dom X & x in X . a ) by CARD_5:2;
J is Subsignature of S1 by Def2;
then ( dom X = the carrier of J & the carrier of J c= the carrier of S1 & the carrier of S1 = dom (X extended_by ({}, the carrier of S1)) ) by ;
then reconsider a = a as SortSymbol of S1 by A1;
assume A2: ( L is subst-correct & L is vf-qc-correct ) ; :: thesis: (\for (x,(A \imp B))) \imp ((\for (x,A)) \imp (\for (x,B))) in G
then (\for (x,(A \imp B))) \imp ((\for (x,A)) \imp B) in G by Th107;
then A3: \for (x,((\for (x,(A \imp B))) \imp ((\for (x,A)) \imp B))) in G by Def39;
A4: ( vf (\for (x,(A \imp B))) = (vf (A \imp B)) (\) () & x in {x} ) by ;
then (vf (\for (x,(A \imp B)))) . a = ((vf (A \imp B)) . a) \ (() . a) by PBOOLE:def 6
.= ((vf (A \imp B)) . a) \ {x} by AOFA_A00:6 ;
then x nin (vf (\for (x,(A \imp B)))) . a by ;
then A5: (\for (x,(A \imp B))) \imp (\for (x,((\for (x,A)) \imp B))) in G by ;
A6: ( vf (\for (x,A)) = (vf A) (\) () & x in {x} ) by ;
then (vf (\for (x,A))) . a = ((vf A) . a) \ (() . a) by PBOOLE:def 6
.= ((vf A) . a) \ {x} by AOFA_A00:6 ;
then x nin (vf (\for (x,A))) . a by ;
then (\for (x,((\for (x,A)) \imp B))) \imp ((\for (x,A)) \imp (\for (x,B))) in G by ;
hence (\for (x,(A \imp B))) \imp ((\for (x,A)) \imp (\for (x,B))) in G by ; :: thesis: verum