Consider the following premises. forall x_9: ~P_7(x_9). forall x_9: P_9(x_9). forall x_9: P_7(x_9) <-> P_9(x_9). Can we infer forall x_9: W(x_9) or Q(x_9) from them? If possible, name the inference rules being used at each step.
 From forall x_9: P_9(x_9) we can infer forall x_9: P_9(x_9) or W(x_9) via universal addition. From forall x_9: P_7(x_9) <-> P_9(x_9), forall x_9: ~P_7(x_9) we can infer forall x_9: ~P_9(x_9) via universal biconditional elimination. From forall x_9: P_9(x_9) or W(x_9), forall x_9: ~P_9(x_9) we can infer forall x_9: W(x_9) via universal disjunctive syllogism. Finally, from forall x_9: W(x_9) we can infer forall x_9: W(x_9) or Q(x_9) via universal addition. Therefore, the answer is yes. Notice, however, that the premises were contradictory, so we can infer anything from them.