Consider the following premises. forall x_5: Q(x_5) and Q_5(x_5). exists x_5: ~Q(x_5). Can we infer exists x_5: P_2(x_5) from them? If possible, name the inference rules being used at each step.
 From forall x_5: Q(x_5) and Q_5(x_5) we can infer forall x_5: Q(x_5) via universal simplification. From forall x_5: Q(x_5) we can infer forall x_5: Q(x_5) or P_2(x_5) via universal addition. Finally, from forall x_5: Q(x_5) or P_2(x_5), exists x_5: ~Q(x_5) we can infer exists x_5: P_2(x_5) via existential disjunctive syllogism. Therefore, the answer is yes. Notice, however, that the premises were contradictory, so we can infer anything from them.