Consider the following premises. forall x: T(x) or ~W_2(x). forall x: T(x) -> W_2(x). exists x: ~W_2(x). Can we infer exists x: ~T(x) from them?
 From forall x: T(x) or ~W_2(x) we can infer forall x: W_2(x) -> T(x). From forall x: W_2(x) -> T(x), forall x: T(x) -> W_2(x) we can infer forall x: W_2(x) <-> T(x). Finally, from forall x: W_2(x) <-> T(x), exists x: ~W_2(x) we can infer exists x: ~T(x). Therefore, the answer is yes.