Consider the following premises. forall x_12: P_7(x_12) or Q_7(x_12). forall x_12: P_12(x_12) and Q_12(x_12). forall x_12: Q_7(x_12) -> R_7(x_12). exists x_12: P_7(x_12) -> R_7(x_12). Can we infer exists x_12: P_12(x_12) and R_7(x_12) from them? If possible, name the inference rules being used at each step.
 From forall x_12: P_12(x_12) and Q_12(x_12) we can infer forall x_12: P_12(x_12) via universal simplification. From exists x_12: P_7(x_12) -> R_7(x_12), forall x_12: Q_7(x_12) -> R_7(x_12), forall x_12: P_7(x_12) or Q_7(x_12) we can infer exists x_12: R_7(x_12) via existential disjunction elimination. Finally, from forall x_12: P_12(x_12), exists x_12: R_7(x_12) we can infer exists x_12: P_12(x_12) and R_7(x_12) via existential conjunction. Therefore, the answer is yes.