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 Th80:
  not x in still_not-bound_in q implies (Ex(x,p) => q) => All(x,p
  => q) is valid & All(x,p => q) => (Ex(x,p) => q) is valid
proof
  assume
A1: not x in still_not-bound_in q;
  p => Ex(x,p) is valid by Th15;
  then
A2: (Ex(x,p) => q) => (p => q) is valid by LUKASI_1:41;
  not x in still_not-bound_in Ex(x,p) by Th6;
  then not x in still_not-bound_in Ex(x,p) => q by A1,Th7;
  hence (Ex(x,p) => q) => All(x,p => q) is valid by A2,CQC_THE1:67;
  All(x,p => q) => (Ex(x,p) => Ex(x,q)) is valid by Th34;
  then
A3: (All(x,p => q) '&' Ex(x,p)) => Ex(x,q) is valid by Th1;
  Ex(x,q) => q is valid by A1,Th20;
  then (All(x,p => q) '&' Ex(x,p)) => q is valid by A3,LUKASI_1:42;
  hence thesis by Th3;
end;
