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;
