Consider the following premises. forall z: S(z). forall z: P_2(z) -> T(z). forall z: T(z) <-> U(z). exists z: U(z) or P_2(z). Can we infer exists z: T(z) and S(z) from them? If possible, name the inference rules being used at each step.
 From forall z: T(z) <-> U(z) we can infer forall z: U(z) -> T(z) via universal biconditional elimination. From forall z: U(z) -> T(z), forall z: P_2(z) -> T(z), exists z: U(z) or P_2(z) we can infer exists z: T(z) via existential disjunction elimination. Finally, from exists z: T(z), forall z: S(z) we can infer exists z: T(z) and S(z) via existential conjunction. Therefore, the answer is yes.