Consider the following premises. forall z: P_4(z) <-> P_1(z). forall z: P_1(z) -> W(z). forall z: P_4(z) or R(z). exists z: R(z) -> W(z). Can we infer exists z: W(z) from them? If possible, name the inference rules being used at each step.
 From forall z: P_4(z) <-> P_1(z) we can infer forall z: P_4(z) -> P_1(z) via universal biconditional elimination. From forall z: P_4(z) -> P_1(z), forall z: P_1(z) -> W(z) we can infer forall z: P_4(z) -> W(z) via universal transitivity. Finally, from forall z: P_4(z) -> W(z), exists z: R(z) -> W(z), forall z: P_4(z) or R(z) we can infer exists z: W(z) via existential disjunction elimination. Therefore, the answer is yes.