Consider the following premises. forall z: W(z) or P_2(z). forall z: U_3(z) -> P_2(z). forall z: P_2(z) -> U_3(z). forall z: W(z) -> U_3(z). Can we infer forall z: U_3(z) from them? If possible, name the inference rules being used at each step.
 From forall z: U_3(z) -> P_2(z), forall z: P_2(z) -> U_3(z) we can infer forall z: U_3(z) <-> P_2(z) via universal biconditional introduction. From forall z: U_3(z) <-> P_2(z) we can infer forall z: P_2(z) -> U_3(z) via universal biconditional elimination. Finally, from forall z: W(z) -> U_3(z), forall z: P_2(z) -> U_3(z), forall z: W(z) or P_2(z) we can infer forall z: U_3(z) via universal disjunction elimination. Therefore, the answer is yes.