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 Th117:
       L is subst-correct vf-qc-correct implies
  \for(x,A\impB)\imp(\for(x,\notB)\imp\for(x,\notA)) in G
  proof assume
A1: L is subst-correct vf-qc-correct;
    (A\impB)\imp(\notB\imp\notA) in G by Th57;
    then
A2: \for(x,A\impB)\imp(\for(x,\notB\imp\notA)) in G by A1,Th115;
    \for(x,\notB\imp\notA)\imp(\for(x,\notB)\imp\for(x,\notA)) in G
    by A1,Th109;
    hence \for(x,A\impB)\imp(\for(x,\notB)\imp\for(x,\notA)) in G by A2,Th45;
  end;
