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