Consider the following premises. forall x_4: ~P_3(x_4). forall x_4: W_1(x_4) <-> P_3(x_4). forall x_4: W_1(x_4). Can we infer forall x_4: P_4(x_4) from them? If possible, name the inference rules being used at each step.
 From forall x_4: W_1(x_4) we can infer forall x_4: W_1(x_4) or P_4(x_4) via universal addition. From forall x_4: W_1(x_4) <-> P_3(x_4), forall x_4: ~P_3(x_4) we can infer forall x_4: ~W_1(x_4) via universal biconditional elimination. Finally, from forall x_4: W_1(x_4) or P_4(x_4), forall x_4: ~W_1(x_4) we can infer forall x_4: P_4(x_4) via universal disjunctive syllogism. Therefore, the answer is yes. Notice, however, that the premises were contradictory, so we can infer anything from them.