Consider the following premises. exists x: T(x) -> W(x). forall x: ~P_4(x). forall x: W(x) -> T(x). forall x: W(x) <-> P_4(x). Can we infer exists x: ~T(x) from them?
 From forall x: W(x) <-> P_4(x), forall x: ~P_4(x) we can infer forall x: ~W(x). From exists x: T(x) -> W(x), forall x: W(x) -> T(x) we can infer exists x: T(x) <-> W(x). Finally, from exists x: T(x) <-> W(x), forall x: ~W(x) we can infer exists x: ~T(x). Therefore, the answer is yes.