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 Th77:
  not x in still_not-bound_in q implies Ex(x,p => q) => (All(x,p)
  => q) is valid
proof
  assume
A1: not x in still_not-bound_in q;
  not x in still_not-bound_in All(x,p) by Th5;
  then not x in still_not-bound_in All(x,p) => q by A1,Th7;
  then
A2: Ex(x,All(x,p) => q) => (All(x,p) => q) is valid by Th20;
  All(x,p) => p is valid by CQC_THE1:66;
  then
A3: All(x,(p => q) => (All(x,p) => q)) is valid by Th23,LUKASI_1:41;
  All(x,(p => q) => (All(x,p) => q)) => (Ex(x,p => q) => Ex(x,All(x,p) =>
  q)) is valid by Th34;
  then Ex(x,p => q) => Ex(x,All(x,p) => q) is valid by A3,CQC_THE1:65;
  hence thesis by A2,LUKASI_1:42;
end;
