reserve A for QC-alphabet;
reserve p, q, r, s, p1, q1 for Element of CQC-WFF(A),
  X, Y, Z, X1, X2 for Subset of CQC-WFF(A),
  h for QC-formula of A,
  x, y for bound_QC-variable of A,
  n for Element of NAT;

theorem
  p = h.x & q = h.y & not x in still_not-bound_in h & not y in
  still_not-bound_in h implies All(x,p) <==> All(y,q)
proof
  assume that
A1: p = h.x and
A2: q = h.y and
A3: not x in still_not-bound_in h and
A4: not y in still_not-bound_in h;
  per cases;
  suppose
    x = y;
    hence thesis by A1,A2;
  end;
  suppose
A5: x <> y;
    then not x in still_not-bound_in q by A2,A3,Th64;
    then
A6: All(y,q) => All(x,p) is valid by A1,A2,A4,CQC_THE2:27;
    not y in still_not-bound_in p by A1,A4,A5,Th64;
    then All(x,p) => All(y,q) is valid by A1,A2,A3,CQC_THE2:27;
    hence thesis by A6,Th50;
  end;
end;
