reserve X,Y for set, x,y,z for object, i,j,n for natural number;
reserve
  n for non empty Nat,
  S for non empty non void n PC-correct PCLangSignature,
  L for language MSAlgebra over S,
  F for PC-theory of L,
  A,B,C,D for Formula of L;
reserve
  J for non empty non void Signature,
  T for non-empty MSAlgebra over J,
  X for non empty-yielding GeneratorSet of T,
  S1 for J-extension non empty non void n PC-correct QC-correct
  QCLangSignature over Union X,
  L for non-empty Language of X extended_by ({},the carrier of S1), S1,
  G for QC-theory of L,
  A,B,C,D for Formula of L;
reserve x,y,z for Element of Union X;
reserve x0,y0,z0 for Element of Union (X extended_by ({},the carrier of S1));
reserve a for SortSymbol of J;

theorem
  L is vf-qc-correct subst-correct implies
  \ex(x,\for(y,A))\imp\for(y,\ex(x,A)) in G
  proof
    assume A1: L is vf-qc-correct subst-correct;
    then A\imp\ex(x,A) in G by Th112;
    then \for(y,A\imp\ex(x,A)) in G &
    \for(y,A\imp\ex(x,A))\imp(\for(y,A)\imp\for(y,\ex(x,A))) in G
    by A1,Def39,Th109;
    then \for(y,A)\imp\for(y,\ex(x,A)) in G by Def38;
    then
A2: \for(x,\for(y,A)\imp\for(y,\ex(x,A))) in G by Def39;
    consider a being object such that
A3: a in dom X & y in X.a by CARD_5:2;
    consider b being object such that
A4: b in dom X & x in X.b by CARD_5:2;
    J is Subsignature of S1 by Def2;
    then dom X = the carrier of J c= the carrier of S1
    by PARTFUN1:def 2,INSTALG1:10;
    then reconsider a,b as Element of S1 by A3,A4;
    reconsider c = b as Element of J by A4;
    vf\for(y,\ex(x,A)) = (vf\ex(x,A))(\)(a-singleton(y)) by A1,A3;
    then (vf\for(y,\ex(x,A))).b = (vf\ex(x,A)).b\(a-singleton(y)).b
    by PBOOLE:def 6;
    then x nin (vf\for(y,\ex(x,A))).c by A1,A4,Th114;
    then \for(x,\for(y,A)\imp\for(y,\ex(x,A)))\imp
    (\ex(x,\for(y,A))\imp\for(y,\ex(x,A))) in G by A1,A4,Th120;
    hence \ex(x,\for(y,A))\imp\for(y,\ex(x,A)) in G by A2,Def38;
  end;
