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 Th25:
  p=h.x & q=h.y & not x in still_not-bound_in h implies All(x,p) => q is valid
proof
  assume that
A1: p=h.x and
A2: q=h.y and
A3: not x in still_not-bound_in h;
A4: (All(x,p) => h).y = (All(x,p).y) => q by A2,Th12
    .= All(x,p) => q by CQC_LANG:27;
  not x in still_not-bound_in All(x,p) by Th5;
  then
A5: All(x,p) => p is valid & not x in still_not-bound_in All(x,p) => h by A3
,Th7,CQC_THE1:66;
  (All(x,p) => h).x = (All(x,p).x) => p by A1,Th12
    .= All(x,p) => p by CQC_LANG:27;
  hence thesis by A4,A5,CQC_THE1:68;
end;
