Consider the following premises. forall x_1: P(x_1) <-> R(x_1). forall x_1: R(x_1) -> ~W_2(x_1). forall x_1: W_2(x_1) and W_1(x_1). forall x_1: R(x_1) -> P(x_1). Can we infer forall x_1: ~P(x_1) from them? If possible, name the inference rules being used at each step.
 From forall x_1: W_2(x_1) and W_1(x_1) we can infer forall x_1: W_2(x_1) via universal simplification. From forall x_1: P(x_1) <-> R(x_1) we can infer forall x_1: P(x_1) -> R(x_1) via universal biconditional elimination. From forall x_1: R(x_1) -> P(x_1), forall x_1: P(x_1) -> R(x_1) we can infer forall x_1: R(x_1) <-> P(x_1) via universal biconditional introduction. From forall x_1: R(x_1) -> ~W_2(x_1), forall x_1: W_2(x_1) we can infer forall x_1: ~R(x_1) via universal modus tollens. Finally, from forall x_1: R(x_1) <-> P(x_1), forall x_1: ~R(x_1) we can infer forall x_1: ~P(x_1) via universal biconditional elimination. Therefore, the answer is yes.