Consider the following premises. forall x_1: W(x_1) -> P(x_1). forall x_1: P(x_1) -> W(x_1). exists x_1: ~W(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: P(x_1) -> W(x_1), forall x_1: W(x_1) -> P(x_1) we can infer forall x_1: P(x_1) <-> W(x_1) via universal biconditional introduction. From forall x_1: P(x_1) <-> W(x_1) we can infer forall x_1: P(x_1) -> W(x_1) via universal biconditional elimination. Finally, from forall x_1: P(x_1) -> W(x_1), exists x_1: ~W(x_1) we can infer exists x_1: ~P(x_1) via existential modus tollens, which contradicts forall x_1: P(x_1). Therefore, the answer is no.