Consider the following premises. exists z_1: P(z_1) -> ~U(z_1). forall z_1: P(z_1) -> T_1(z_1). forall z_1: U(z_1). forall z_1: T_1(z_1) <-> P(z_1). Can we infer exists z_1: ~T_1(z_1) from them? If possible, name the inference rules being used at each step.
 From forall z_1: T_1(z_1) <-> P(z_1) we can infer forall z_1: T_1(z_1) -> P(z_1) via universal biconditional elimination. From exists z_1: P(z_1) -> ~U(z_1), forall z_1: U(z_1) we can infer exists z_1: ~P(z_1) via existential modus tollens. From forall z_1: T_1(z_1) -> P(z_1), forall z_1: P(z_1) -> T_1(z_1) we can infer forall z_1: T_1(z_1) <-> P(z_1) via universal biconditional introduction. Finally, from forall z_1: T_1(z_1) <-> P(z_1), exists z_1: ~P(z_1) we can infer exists z_1: ~T_1(z_1) via existential biconditional elimination. Therefore, the answer is yes.