reserve i,k for Nat;
reserve A for QC-alphabet;
reserve x for bound_QC-variable of A;
reserve a for free_QC-variable of A;
reserve p,q for Element of QC-WFF(A);
reserve l for FinSequence of QC-variables(A);
reserve P,Q for QC-pred_symbol of A;
reserve V for non empty Subset of QC-variables(A);
reserve s,t for QC-symbol of A;

theorem Th23:
  for p being QC-formula of A holds All(x,p) is closed iff
  still_not-bound_in p c= {x}
proof
  let p be QC-formula of A;
  thus All(x,p) is closed implies still_not-bound_in p c= {x}
  proof
    assume still_not-bound_in All(x,p) = {};
    then {} = (still_not-bound_in p) \ {x} by Th12;
    hence thesis by XBOOLE_1:37;
  end;
  assume still_not-bound_in p c= {x};
  then {} = (still_not-bound_in p) \ {x} by XBOOLE_1:37;
  hence still_not-bound_in All(x,p) = {} by Th12;
end;
