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,y,A)\imp\ex(y,x,A) in G
  proof
    assume A1: L is vf-qc-correct subst-correct;
    then \for(y,x,\notA)\imp\for(x,y,\notA) in G by Th138;
    then
A2: \not\for(x,y,\notA)\imp\not\for(y,x,\notA) in G by Th58;
    \ex(x,y,A)\iff\not\for(x,y,\notA) in G &
    \ex(y,x,A)\iff\not\for(y,x,\notA) in G by A1,Th111;
    then \ex(x,y,A)\imp\not\for(y,x,\notA) in G &
    \not\for(y,x,\notA)\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;
  end;
