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 Th122:
       L is subst-correct vf-qc-correct implies
  \for(x,\notA)\iff\not\ex(x,A) in G
  proof assume
A1: L is subst-correct vf-qc-correct;
    A\imp\not\notA in G by Th64;
    then \for(x,A\imp\not\notA) in G &
    \for(x,A\imp\not\notA)\imp(\ex(x,A)\imp\ex(x,\not\notA)) in G
    by A1,Def39,Th121;
    then
A2: \ex(x,A)\imp\ex(x,\not\notA) in G by Def38;
    \not\notA\impA in G by Th65;
    then \for(x,\not\notA\impA) in G &
    \for(x,\not\notA\impA)\imp(\ex(x,\not\notA)\imp\ex(x,A)) in G
    by A1,Def39,Th121;
    then \ex(x,\not\notA)\imp\ex(x,A) in G by Def38;
    then \ex(x,A)\iff\ex(x,\not\notA) in G &
    \ex(x,\not\notA)\iff\not\for(x,\notA) in G by Def39,A2,Th43; then
    \ex(x,A)\iff\not\for(x,\notA) in G by Th91; then
    \not\for(x,\notA)\iff\ex(x,A) in G by Th90; then
    \not\not\for(x,\notA)\iff\not\ex(x,A) in G by Th94;
    hence \for(x,\notA)\iff\not\ex(x,A) in G by Th95;
  end;
