Consider the following premises. exists y_2: T(y_2) -> S_2(y_2). forall y_2: P(y_2) -> T(y_2). forall y_2: S_2(y_2) or ~W_1(y_2). forall y_2: S_2(y_2) -> W_1(y_2). Can we infer exists y_2: P(y_2) -> W_1(y_2) from them? If possible, name the inference rules being used at each step.
 From forall y_2: P(y_2) -> T(y_2), exists y_2: T(y_2) -> S_2(y_2) we can infer exists y_2: P(y_2) -> S_2(y_2) via existential transitivity. From forall y_2: S_2(y_2) or ~W_1(y_2) we can infer forall y_2: W_1(y_2) -> S_2(y_2) via universal rewriting. From forall y_2: S_2(y_2) -> W_1(y_2), forall y_2: W_1(y_2) -> S_2(y_2) we can infer forall y_2: S_2(y_2) <-> W_1(y_2) via universal biconditional introduction. From forall y_2: S_2(y_2) <-> W_1(y_2) we can infer forall y_2: S_2(y_2) -> W_1(y_2) via universal biconditional elimination. Finally, from exists y_2: P(y_2) -> S_2(y_2), forall y_2: S_2(y_2) -> W_1(y_2) we can infer exists y_2: P(y_2) -> W_1(y_2) via existential transitivity. Therefore, the answer is yes.