Consider the following premises. forall x_12: ~P_2(x_12). forall x_12: P_12(x_12) or Q_12(x_12). exists x_12: Q_12(x_12) -> P(x_12). forall x_12: P_2(x_12) or U(x_12). forall x_12: P_12(x_12) -> P(x_12). Can we infer exists x_12: U(x_12) and P(x_12) from them? If possible, name the inference rules being used at each step.
 From forall x_12: P_12(x_12) -> P(x_12), exists x_12: Q_12(x_12) -> P(x_12), forall x_12: P_12(x_12) or Q_12(x_12) we can infer exists x_12: P(x_12) via existential disjunction elimination. From forall x_12: P_2(x_12) or U(x_12), forall x_12: ~P_2(x_12) we can infer forall x_12: U(x_12) via universal disjunctive syllogism. Finally, from forall x_12: U(x_12), exists x_12: P(x_12) we can infer exists x_12: U(x_12) and P(x_12) via existential conjunction. Therefore, the answer is yes.