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 Th22:
  p=h.x & q=h.y & not y in still_not-bound_in h implies p => Ex(y, q) is valid
proof
  assume that
A1: p=h.x and
A2: q=h.y and
A3: not y in still_not-bound_in h;
A4: (h => Ex(y,q)).x = (h.x) => (Ex(y,q).x) by Th12
    .= p => Ex(y,q) by A1,CQC_LANG:27;
  not y in still_not-bound_in Ex(y,q) by Th6;
  then
A5: not y in still_not-bound_in h => Ex(y,q) by A3,Th7;
A6: q => Ex(y,q) is valid by Th15;
  (h => Ex(y,q)).y = (h.y) => (Ex(y,q).y) by Th12
    .= q => Ex(y,q) by A2,CQC_LANG:27;
  hence thesis by A6,A4,A5,CQC_THE1:68;
end;
