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

(* Specifications: *)
Lemma notCon2wNotCon : forall T : System, Inconsistent LNN T -> wInconsistent T.