Consider the following premises. forall x_5: Q_5(x_5) -> R_5(x_5). exists x_5: Q_1(x_5) or U(x_5). forall x_5: Q_1(x_5) -> Q_5(x_5). forall x_5: ~R_5(x_5). Can we infer exists x_5: U(x_5) from them? If possible, name the inference rules being used at each step.
 From forall x_5: Q_1(x_5) -> Q_5(x_5), forall x_5: Q_5(x_5) -> R_5(x_5) we can infer forall x_5: Q_1(x_5) -> R_5(x_5) via universal transitivity. From forall x_5: Q_1(x_5) -> R_5(x_5), forall x_5: ~R_5(x_5) we can infer forall x_5: ~Q_1(x_5) via universal modus tollens. Finally, from exists x_5: Q_1(x_5) or U(x_5), forall x_5: ~Q_1(x_5) we can infer exists x_5: U(x_5) via existential disjunctive syllogism. Therefore, the answer is yes.