Consider the following premises. forall x_18: Q_18(x_18) -> R_18(x_18). forall x_18: P_18(x_18) or Q_5(x_18). exists x_18: P_18(x_18) -> Q_18(x_18). forall x_18: ~R_18(x_18). Can we infer exists x_18: Q_5(x_18) or Q(x_18) from them? If possible, name the inference rules being used at each step.
 From exists x_18: P_18(x_18) -> Q_18(x_18), forall x_18: Q_18(x_18) -> R_18(x_18) we can infer exists x_18: P_18(x_18) -> R_18(x_18) via existential transitivity. From exists x_18: P_18(x_18) -> R_18(x_18), forall x_18: ~R_18(x_18) we can infer exists x_18: ~P_18(x_18) via existential modus tollens. From forall x_18: P_18(x_18) or Q_5(x_18), exists x_18: ~P_18(x_18) we can infer exists x_18: Q_5(x_18) via existential disjunctive syllogism. Finally, from exists x_18: Q_5(x_18) we can infer exists x_18: Q_5(x_18) or Q(x_18) via existential addition. Therefore, the answer is yes.