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