Consider the following premises. forall z: T(z) <-> T_2(z). exists z: T_1(z) <-> T(z). forall z: ~T_2(z). Can we infer exists z: ~T_1(z) from them?
 From exists z: T_1(z) <-> T(z) we can infer exists z: T_1(z) -> T(z). From forall z: T(z) <-> T_2(z), forall z: ~T_2(z) we can infer forall z: ~T(z). Finally, from exists z: T_1(z) -> T(z), forall z: ~T(z) we can infer exists z: ~T_1(z). Therefore, the answer is yes.