Consider the following premises. forall y_2: S_2(y_2) or ~W_1(y_2). forall y_2: S_2(y_2) -> W_1(y_2). exists y_2: T(y_2) -> S_2(y_2). forall y_2: P(y_2) -> T(y_2). Can we infer exists y_2: P(y_2) -> W_1(y_2) from them?
 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). 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). 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). 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). 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). Therefore, the answer is yes.