Consider the following premises. forall x_12: P_8(x_12) -> P_12(x_12). exists x_12: P_12(x_12) or ~Q_12(x_12). forall x_12: P_8(x_12) or Q_12(x_12). Can we infer exists x_12: P_12(x_12) or Q(x_12) from them? If possible, name the inference rules being used at each step.
 From exists x_12: P_12(x_12) or ~Q_12(x_12) we can infer exists x_12: Q_12(x_12) -> P_12(x_12) via existential rewriting. From forall x_12: P_8(x_12) -> P_12(x_12), exists x_12: Q_12(x_12) -> P_12(x_12), forall x_12: P_8(x_12) or Q_12(x_12) we can infer exists x_12: P_12(x_12) via existential disjunction elimination. Finally, from exists x_12: P_12(x_12) we can infer exists x_12: P_12(x_12) or Q(x_12) via existential addition. Therefore, the answer is yes.