Consider the following premises. forall z_4: W(z_4) -> S(z_4). forall z_4: U_1(z_4) -> W(z_4). exists z_4: U_1(z_4) or ~S(z_4). Can we infer exists z_4: U_1(z_4) -> S(z_4) from them? If possible, name the inference rules being used at each step.
 From forall z_4: U_1(z_4) -> W(z_4), forall z_4: W(z_4) -> S(z_4) we can infer forall z_4: U_1(z_4) -> S(z_4) via universal transitivity. From exists z_4: U_1(z_4) or ~S(z_4) we can infer exists z_4: S(z_4) -> U_1(z_4) via existential rewriting. From forall z_4: U_1(z_4) -> S(z_4), exists z_4: S(z_4) -> U_1(z_4) we can infer exists z_4: U_1(z_4) <-> S(z_4) via existential biconditional introduction. Finally, from exists z_4: U_1(z_4) <-> S(z_4) we can infer exists z_4: U_1(z_4) -> S(z_4) via existential biconditional elimination. Therefore, the answer is yes.