Consider the following premises. exists z_2: Q_3(z_2) -> P_1(z_2). forall z_2: Q_3(z_2) or ~T_2(z_2). forall z_2: P_1(z_2) -> T_2(z_2). forall z_2: ~Q_3(z_2). Can we infer exists z_2: ~T_2(z_2) from them? If possible, name the inference rules being used at each step.
 From exists z_2: Q_3(z_2) -> P_1(z_2), forall z_2: P_1(z_2) -> T_2(z_2) we can infer exists z_2: Q_3(z_2) -> T_2(z_2) via existential transitivity. From forall z_2: Q_3(z_2) or ~T_2(z_2) we can infer forall z_2: T_2(z_2) -> Q_3(z_2) via universal rewriting. From exists z_2: Q_3(z_2) -> T_2(z_2), forall z_2: T_2(z_2) -> Q_3(z_2) we can infer exists z_2: Q_3(z_2) <-> T_2(z_2) via existential biconditional introduction. Finally, from exists z_2: Q_3(z_2) <-> T_2(z_2), forall z_2: ~Q_3(z_2) we can infer exists z_2: ~T_2(z_2) via existential biconditional elimination. Therefore, the answer is yes.