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