reserve A for QC-alphabet;
reserve X,T for Subset of CQC-WFF(A);
reserve F,G,H,p,q,r,t for Element of CQC-WFF(A);
reserve s,h for QC-formula of A;
reserve x,y for bound_QC-variable of A;
reserve f for FinSequence of [:CQC-WFF(A),Proof_Step_Kinds:];
reserve i,j for Element of NAT;

theorem
  not x in still_not-bound_in q implies ( All(x,p) => q is valid iff Ex(
  x,p => q ) is valid )
proof
  assume not x in still_not-bound_in q;
  then
A1: Ex(x,p => q) => (All(x,p) => q ) is valid by Th77;
  (All(x,p) => q) => Ex(x,p => q) is valid by Th78;
  then (All(x,p) => q) <=> Ex(x,p => q) is valid by A1,Lm14;
  hence thesis by Lm15;
end;
