Consider the following premises. forall y_1: P(y_1) -> W(y_1). s. W(j_2) -> ~s. forall y_1: W(y_1) -> P(y_1). Can we infer ~P(j_2) from them? If possible, name the inference rules being used at each step.
 From forall y_1: P(y_1) -> W(y_1), forall y_1: W(y_1) -> P(y_1) we can infer forall y_1: P(y_1) <-> W(y_1) via universal biconditional introduction. From forall y_1: P(y_1) <-> W(y_1) we can infer P(j_2) <-> W(j_2) via universal instantiation. From W(j_2) -> ~s, s we can infer ~W(j_2) via modus tollens. Finally, from P(j_2) <-> W(j_2), ~W(j_2) we can infer ~P(j_2) via biconditional elimination. Therefore, the answer is yes.