Consider the following premises. exists z: T_3(z) -> P_1(z). forall z: ~P(z). forall z: T_3(z) <-> P(z). forall z: P_1(z) -> T_3(z). Can we infer exists z: ~P_1(z) from them? If possible, name the inference rules being used at each step.
 From forall z: T_3(z) <-> P(z), forall z: ~P(z) we can infer forall z: ~T_3(z) via universal biconditional elimination. From forall z: P_1(z) -> T_3(z), exists z: T_3(z) -> P_1(z) we can infer exists z: P_1(z) <-> T_3(z) via existential biconditional introduction. Finally, from exists z: P_1(z) <-> T_3(z), forall z: ~T_3(z) we can infer exists z: ~P_1(z) via existential biconditional elimination. Therefore, the answer is yes.