Consider the following premises. exists y: W_3(y) -> W(y). forall y: S(y) -> W(y). forall y: W(y) -> S(y). forall y: W(y) -> W_3(y). Can we infer exists y: S(y) -> W_3(y) from them? If possible, name the inference rules being used at each step.
 From forall y: S(y) -> W(y), forall y: W(y) -> S(y) we can infer forall y: S(y) <-> W(y) via universal biconditional introduction. From forall y: W(y) -> W_3(y), exists y: W_3(y) -> W(y) we can infer exists y: W(y) <-> W_3(y) via existential biconditional introduction. From exists y: W(y) <-> W_3(y) we can infer exists y: W(y) -> W_3(y) via existential biconditional elimination. From forall y: S(y) <-> W(y) we can infer forall y: S(y) -> W(y) via universal biconditional elimination. Finally, from forall y: S(y) -> W(y), exists y: W(y) -> W_3(y) we can infer exists y: S(y) -> W_3(y) via existential transitivity. Therefore, the answer is yes.