Consider the following premises. forall x: R_2(x) -> S_1(x). exists x: S_1(x) -> R_2(x). forall x: R_2(x) or ~W(x). Can we infer exists x: W(x) -> S_1(x) from them? If possible, name the inference rules being used at each step.
 From forall x: R_2(x) -> S_1(x), exists x: S_1(x) -> R_2(x) we can infer exists x: R_2(x) <-> S_1(x) via existential biconditional introduction. From exists x: R_2(x) <-> S_1(x) we can infer exists x: R_2(x) -> S_1(x) via existential biconditional elimination. From forall x: R_2(x) or ~W(x) we can infer forall x: W(x) -> R_2(x) via universal rewriting. Finally, from forall x: W(x) -> R_2(x), exists x: R_2(x) -> S_1(x) we can infer exists x: W(x) -> S_1(x) via existential transitivity. Therefore, the answer is yes.