Consider the following premises. exists x_9: Q_9(x_9). forall x_9: Q(x_9) -> P(x_9). forall x_9: P(x_9) -> Q(x_9). forall x_9: Q(x_9) -> ~Q_9(x_9). Can we infer exists x_9: ~P(x_9) from them? If possible, name the inference rules being used at each step.
 From forall x_9: Q(x_9) -> ~Q_9(x_9), exists x_9: Q_9(x_9) we can infer exists x_9: ~Q(x_9) via existential modus tollens. From forall x_9: Q(x_9) -> P(x_9), forall x_9: P(x_9) -> Q(x_9) we can infer forall x_9: Q(x_9) <-> P(x_9) via universal biconditional introduction. From forall x_9: Q(x_9) <-> P(x_9) we can infer forall x_9: P(x_9) -> Q(x_9) via universal biconditional elimination. Finally, from forall x_9: P(x_9) -> Q(x_9), exists x_9: ~Q(x_9) we can infer exists x_9: ~P(x_9) via existential modus tollens. Therefore, the answer is yes.