Consider the following premises. forall x_17: Q_4(x_17) and Q_17(x_17). exists x_17: Q_4(x_17) -> ~P_13(x_17). forall x_17: P_13(x_17) and Q_13(x_17). Can we infer exists x_17: Q(x_17) from them? If possible, name the inference rules being used at each step.
 From forall x_17: Q_4(x_17) and Q_17(x_17) we can infer forall x_17: Q_4(x_17) via universal simplification. From forall x_17: P_13(x_17) and Q_13(x_17) we can infer forall x_17: P_13(x_17) via universal simplification. From exists x_17: Q_4(x_17) -> ~P_13(x_17), forall x_17: P_13(x_17) we can infer exists x_17: ~Q_4(x_17) via existential modus tollens. From forall x_17: Q_4(x_17) we can infer forall x_17: Q_4(x_17) or Q(x_17) via universal addition. Finally, from forall x_17: Q_4(x_17) or Q(x_17), exists x_17: ~Q_4(x_17) we can infer exists x_17: Q(x_17) via existential disjunctive syllogism. Therefore, the answer is yes. Notice, however, that the premises were contradictory, so we can infer anything from them.