Consider the following premises. forall x_9: Q_9(x_9) -> P(x_9). forall x_9: P_9(x_9) -> P(x_9). forall x_9: Q_5(x_9). forall x_9: P_9(x_9) or Q_9(x_9). Can we infer exists x_9: ~(P(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) -> P(x_9), forall x_9: Q_9(x_9) -> P(x_9), forall x_9: P_9(x_9) or Q_9(x_9) we can infer forall x_9: P(x_9) via universal disjunction elimination. From forall x_9: P(x_9), forall x_9: Q_5(x_9) we can infer forall x_9: P(x_9) and Q_5(x_9) via universal conjunction. From forall x_9: P(x_9) and Q_5(x_9) we can infer forall x_9: P(x_9) via universal simplification. Finally, from forall x_9: P(x_9) we can infer forall x_9: P(x_9) or Q(x_9) via universal addition, which contradicts exists x_9: ~(P(x_9) or Q(x_9)). Therefore, the answer is no.