Consider the following premises. forall z: P_3(z) <-> U_2(z). forall z: S(z) -> W_2(z). forall z: U_2(z) -> U(z). exists z: U(z) -> S(z). Can we infer exists z: P_3(z) -> W_2(z) from them? If possible, name the inference rules being used at each step.
 From forall z: P_3(z) <-> U_2(z) we can infer forall z: P_3(z) -> U_2(z) via universal biconditional elimination. From forall z: P_3(z) -> U_2(z), forall z: U_2(z) -> U(z) we can infer forall z: P_3(z) -> U(z) via universal transitivity. From exists z: U(z) -> S(z), forall z: S(z) -> W_2(z) we can infer exists z: U(z) -> W_2(z) via existential transitivity. Finally, from forall z: P_3(z) -> U(z), exists z: U(z) -> W_2(z) we can infer exists z: P_3(z) -> W_2(z) via existential transitivity. Therefore, the answer is yes.