Consider the following premises. forall y_2: P_3(y_2) -> S_1(y_2). forall y_2: P_1(y_2) <-> P_3(y_2). forall y_2: T_2(y_2) <-> S_1(y_2). forall y_2: ~T_2(y_2). forall y_2: S_1(y_2) -> P_3(y_2). Can we infer forall y_2: ~P_1(y_2) from them? If possible, name the inference rules being used at each step.
 From forall y_2: S_1(y_2) -> P_3(y_2), forall y_2: P_3(y_2) -> S_1(y_2) we can infer forall y_2: S_1(y_2) <-> P_3(y_2) via universal biconditional introduction. From forall y_2: T_2(y_2) <-> S_1(y_2) we can infer forall y_2: S_1(y_2) -> T_2(y_2) via universal biconditional elimination. From forall y_2: S_1(y_2) -> T_2(y_2), forall y_2: ~T_2(y_2) we can infer forall y_2: ~S_1(y_2) via universal modus tollens. From forall y_2: S_1(y_2) <-> P_3(y_2), forall y_2: ~S_1(y_2) we can infer forall y_2: ~P_3(y_2) via universal biconditional elimination. Finally, from forall y_2: P_1(y_2) <-> P_3(y_2), forall y_2: ~P_3(y_2) we can infer forall y_2: ~P_1(y_2) via universal biconditional elimination. Therefore, the answer is yes.