Consider the following premises. exists x_9: P_4(x_9) and S(x_9). forall x_9: ~P_4(x_9). Can we infer exists x_9: Q_3(x_9) from them? If possible, name the inference rules being used at each step.
 From exists x_9: P_4(x_9) and S(x_9) we can infer exists x_9: P_4(x_9) via existential simplification. From exists x_9: P_4(x_9) we can infer exists x_9: P_4(x_9) or Q_3(x_9) via existential addition. Finally, from exists x_9: P_4(x_9) or Q_3(x_9), forall x_9: ~P_4(x_9) we can infer exists x_9: Q_3(x_9) via existential disjunctive syllogism. Therefore, the answer is yes. Notice, however, that the premises were contradictory, so we can infer anything from them.