Consider the following premises. forall x_1: P_1(x_1) -> P_3(x_1). forall x_1: P_1(x_1) -> W_2(x_1). forall x_1: W_2(x_1) -> P_1(x_1). Can we infer forall x_1: W_2(x_1) -> P_3(x_1) from them? If possible, name the inference rules being used at each step.
 From forall x_1: W_2(x_1) -> P_1(x_1), forall x_1: P_1(x_1) -> W_2(x_1) we can infer forall x_1: W_2(x_1) <-> P_1(x_1) via universal biconditional introduction. From forall x_1: W_2(x_1) <-> P_1(x_1) we can infer forall x_1: W_2(x_1) -> P_1(x_1) via universal biconditional elimination. Finally, from forall x_1: W_2(x_1) -> P_1(x_1), forall x_1: P_1(x_1) -> P_3(x_1) we can infer forall x_1: W_2(x_1) -> P_3(x_1) via universal transitivity. Therefore, the answer is yes.