Consider the following premises. forall x_7: P_5(x_7). forall x_7: ~P_7(x_7). forall x_7: P_7(x_7) or Q_7(x_7). Can we infer exists x_7: ~(P_5(x_7) or Q(x_7)) from them? If possible, name the inference rules being used at each step.
 From forall x_7: P_7(x_7) or Q_7(x_7), forall x_7: ~P_7(x_7) we can infer forall x_7: Q_7(x_7) via universal disjunctive syllogism. From forall x_7: P_5(x_7), forall x_7: Q_7(x_7) we can infer forall x_7: P_5(x_7) and Q_7(x_7) via universal conjunction. From forall x_7: P_5(x_7) and Q_7(x_7) we can infer forall x_7: P_5(x_7) via universal simplification. Finally, from forall x_7: P_5(x_7) we can infer forall x_7: P_5(x_7) or Q(x_7) via universal addition, which contradicts exists x_7: ~(P_5(x_7) or Q(x_7)). Therefore, the answer is no.