Consider the following premises. forall x_1: U(x_1) or Q(x_1). exists x_1: U(x_1) -> Q_1(x_1). forall x_1: ~Q_1(x_1). Can we infer exists x_1: Q(x_1) from them? If possible, name the inference rules being used at each step.
 From exists x_1: U(x_1) -> Q_1(x_1), forall x_1: ~Q_1(x_1) we can infer exists x_1: ~U(x_1) via existential modus tollens. Finally, from forall x_1: U(x_1) or Q(x_1), exists x_1: ~U(x_1) we can infer exists x_1: Q(x_1) via existential disjunctive syllogism. Therefore, the answer is yes.