Definition wConsistent (T : System) :=
  forall (f : Formula) (v : nat),
  (forall x : nat, In x (freeVarFormula LNN f) -> v = x) ->
  SysPrf T (existH v (notH f)) ->
  exists n : nat, ~ SysPrf T (substituteFormula LNN f v (natToTerm n)).

(* Specifications: *)
Lemma wCon2Con : forall T : System, wConsistent T -> Consistent LNN T.