Consider the following premises. forall z: Q(z) <-> P(z). forall z: Q(z) or T_1(z). forall z: P(z) -> T_1(z). forall z: T_1(z) -> P(z). Can we infer exists z: ~P(z) from them?
 From forall z: P(z) -> T_1(z), forall z: T_1(z) -> P(z) we can infer forall z: P(z) <-> T_1(z). From forall z: P(z) <-> T_1(z) we can infer forall z: T_1(z) -> P(z). From forall z: Q(z) <-> P(z) we can infer forall z: Q(z) -> P(z). Finally, from forall z: Q(z) -> P(z), forall z: T_1(z) -> P(z), forall z: Q(z) or T_1(z) we can infer forall z: P(z), which contradicts exists z: ~P(z). Therefore, the answer is no.