Consider the following premises. exists z: T(z) <-> P_2(z). forall z: ~T(z). forall z: Q(z) -> P_2(z). Can we infer exists z: ~Q(z) from them?
 From exists z: T(z) <-> P_2(z) we can infer exists z: P_2(z) -> T(z). From exists z: P_2(z) -> T(z), forall z: ~T(z) we can infer exists z: ~P_2(z). Finally, from forall z: Q(z) -> P_2(z), exists z: ~P_2(z) we can infer exists z: ~Q(z). Therefore, the answer is yes.