Consider the following premises. forall y_4: S_1(y_4) -> W(y_4). exists y_4: W(y_4) -> S(y_4). forall y_4: S(y_4) <-> S_1(y_4). Can we infer exists y_4: S_1(y_4) <-> S(y_4) from them? If possible, name the inference rules being used at each step.
 From forall y_4: S_1(y_4) -> W(y_4), exists y_4: W(y_4) -> S(y_4) we can infer exists y_4: S_1(y_4) -> S(y_4) via existential transitivity. From forall y_4: S(y_4) <-> S_1(y_4) we can infer forall y_4: S(y_4) -> S_1(y_4) via universal biconditional elimination. Finally, from exists y_4: S_1(y_4) -> S(y_4), forall y_4: S(y_4) -> S_1(y_4) we can infer exists y_4: S_1(y_4) <-> S(y_4) via existential biconditional introduction. Therefore, the answer is yes.