Consider the following premises. forall x_9: Q(x_9) or ~U_1(x_9). exists x_9: P_9(x_9) <-> Q(x_9). forall x_9: ~P_9(x_9). Can we infer exists x_9: ~U_1(x_9) from them? If possible, name the inference rules being used at each step.
 From exists x_9: P_9(x_9) <-> Q(x_9), forall x_9: ~P_9(x_9) we can infer exists x_9: ~Q(x_9) via existential biconditional elimination. From forall x_9: Q(x_9) or ~U_1(x_9) we can infer forall x_9: U_1(x_9) -> Q(x_9) via universal rewriting. Finally, from forall x_9: U_1(x_9) -> Q(x_9), exists x_9: ~Q(x_9) we can infer exists x_9: ~U_1(x_9) via existential modus tollens. Therefore, the answer is yes.