Consider the following premises. forall x_7: ~U(x_7). forall x_7: U(x_7) or S(x_7). exists x_7: Q_1(x_7) or ~S(x_7). Can we infer exists x_7: Q_1(x_7) from them? If possible, name the inference rules being used at each step.
 From exists x_7: Q_1(x_7) or ~S(x_7) we can infer exists x_7: S(x_7) -> Q_1(x_7) via existential rewriting. From forall x_7: U(x_7) or S(x_7), forall x_7: ~U(x_7) we can infer forall x_7: S(x_7) via universal disjunctive syllogism. Finally, from exists x_7: S(x_7) -> Q_1(x_7), forall x_7: S(x_7) we can infer exists x_7: Q_1(x_7) via existential modus ponens. Therefore, the answer is yes.