Consider the following premises. forall x_10: W(x_10) -> Q_3(x_10). forall x_10: Q_10(x_10) -> Q_3(x_10). forall x_10: P_2(x_10). forall x_10: S_3(x_10) or W(x_10). exists x_10: S_3(x_10) -> Q_10(x_10). Can we infer exists x_10: P_2(x_10) and Q_3(x_10) from them? If possible, name the inference rules being used at each step.
 From exists x_10: S_3(x_10) -> Q_10(x_10), forall x_10: Q_10(x_10) -> Q_3(x_10) we can infer exists x_10: S_3(x_10) -> Q_3(x_10) via existential transitivity. From exists x_10: S_3(x_10) -> Q_3(x_10), forall x_10: W(x_10) -> Q_3(x_10), forall x_10: S_3(x_10) or W(x_10) we can infer exists x_10: Q_3(x_10) via existential disjunction elimination. Finally, from forall x_10: P_2(x_10), exists x_10: Q_3(x_10) we can infer exists x_10: P_2(x_10) and Q_3(x_10) via existential conjunction. Therefore, the answer is yes.