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