Consider the following premises. forall z: S_3(z) -> U(z). forall z: U_4(z) -> U(z). forall z: P_4(z). exists z: S_3(z) or U_4(z). Can we infer exists z: P_4(z) and U(z) from them? If possible, name the inference rules being used at each step.
 From forall z: S_3(z) -> U(z), forall z: U_4(z) -> U(z), exists z: S_3(z) or U_4(z) we can infer exists z: U(z) via existential disjunction elimination. Finally, from forall z: P_4(z), exists z: U(z) we can infer exists z: P_4(z) and U(z) via existential conjunction. Therefore, the answer is yes.