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));

theorem Th129:
  L is subst-correct vf-qc-correct &
  A\iffB in G implies \ex(x,A)\iff\ex(x,B) in G
  proof assume
A1: L is subst-correct vf-qc-correct;
    assume
A2: A\iffB in G;
    (A\iffB)\imp(A\impB) in G & (A\iffB)\imp(B\impA) in G by Def38;
    then A\impB in G & B\impA in G by A2,Def38;
    then \ex(x,A)\imp\ex(x,B) in G & \ex(x,B)\imp\ex(x,A) in G by A1,Th128;
    then
A3: (\ex(x,A)\imp\ex(x,B))\and(\ex(x,B)\imp\ex(x,A)) in G by Th35;
    (\ex(x,A)\imp\ex(x,B))\and(\ex(x,B)\imp\ex(x,A))\imp(\ex(x,A)\iff\ex(x,B))
    in G by Def38;
    hence thesis by A3,Def38;
  end;
