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 Th83:
  not x in still_not-bound_in p implies Ex(x,p => q) => (p => Ex(x
  ,q)) is valid
proof
  assume
A1: not x in still_not-bound_in p;
  not x in still_not-bound_in Ex(x,q) by Th6;
  then not x in still_not-bound_in p => Ex(x,q) by A1,Th7;
  then
A2: Ex(x,p => Ex(x,q)) => (p => Ex(x,q)) is valid by Th20;
  q => Ex(x,q) is valid by Th15;
  then
A3: All(x,(p => q) => (p => Ex(x,q))) is valid by Th23,LUKASI_1:51;
  All(x,(p => q) => (p => Ex(x,q))) => (Ex(x,p => q) => Ex(x,p => Ex(x,q))
  ) is valid by Th34;
  then (Ex(x,p => q) => Ex(x,p => Ex(x,q))) is valid by A3,CQC_THE1:65;
  hence thesis by A2,LUKASI_1:42;
end;
