Consider the following premises. exists z: W_2(z) or ~U_1(z). forall z: S(z) <-> U_1(z). forall z: W(z) -> S(z). Can we infer exists z: W(z) -> W_2(z) from them? If possible, name the inference rules being used at each step.
 From forall z: S(z) <-> U_1(z) we can infer forall z: S(z) -> U_1(z) via universal biconditional elimination. From exists z: W_2(z) or ~U_1(z) we can infer exists z: U_1(z) -> W_2(z) via existential rewriting. From forall z: S(z) -> U_1(z), exists z: U_1(z) -> W_2(z) we can infer exists z: S(z) -> W_2(z) via existential transitivity. Finally, from forall z: W(z) -> S(z), exists z: S(z) -> W_2(z) we can infer exists z: W(z) -> W_2(z) via existential transitivity. Therefore, the answer is yes.