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 Th120:
  L is subst-correct vf-qc-correct implies
  for a being SortSymbol of J
  st x in X.a & x nin (vf B).a holds \for(x,A\impB)\imp(\ex(x,A)\impB) in G
  proof set Y = X extended_by ({},the carrier of S1);
    assume
A1: L is subst-correct vf-qc-correct;
    let a be SortSymbol of J;assume
A2: x in X.a & x nin (vf B).a;
A3: (A\impB)\imp(\notB\imp\notA) in G by Th57;
A4: \for(x,A\impB)\imp\for(x,\notB\imp\notA) in G by A1,A3,Th115;
    x nin (vf \notB).a by A1,A2;
    then \for(x,\notB\imp\notA)\imp(\notB\imp\for(x,\notA)) in G by A2,Def39;
    then
A5: \for(x,A\impB)\imp(\notB\imp\for(x,\notA)) in G by A4,Th45;
    \not\ex(x,A)\iff\for(x,\notA) in G &
    \not\ex(x,A)\iff\for(x,\notA)\imp
    (\for(x,\notA)\imp\not\ex(x,A)) in G by Def39,Def38;
    then \for(x,\notA)\imp\not\ex(x,A) in G by Def38;
    then
A6: \for(x,A\impB)\imp(\notB\imp\not\ex(x,A)) in G by A5,Th69;
    (\notB\imp\not\ex(x,A))\imp(\ex(x,A)\impB) in G by Def38;
    hence thesis by A6,Th45;
  end;
