Consider the following premises. exists x_9: Q_9(x_9) -> R_9(x_9). forall x_9: P_9(x_9) -> Q_9(x_9). forall x_9: ~P_7(x_9) or R_7(x_9). forall x_9: P_7(x_9) or P_4(x_9). forall x_9: R_7(x_9) -> R_9(x_9). forall x_9: P_4(x_9) -> P_9(x_9). Can we infer exists x_9: R_9(x_9) from them? If possible, name the inference rules being used at each step.
 From forall x_9: P_9(x_9) -> Q_9(x_9), exists x_9: Q_9(x_9) -> R_9(x_9) we can infer exists x_9: P_9(x_9) -> R_9(x_9) via existential transitivity. From forall x_9: P_7(x_9) or P_4(x_9), forall x_9: ~P_7(x_9) or R_7(x_9) we can infer forall x_9: P_4(x_9) or R_7(x_9) via universal resolution. From forall x_9: P_4(x_9) -> P_9(x_9), exists x_9: P_9(x_9) -> R_9(x_9) we can infer exists x_9: P_4(x_9) -> R_9(x_9) via existential transitivity. Finally, from exists x_9: P_4(x_9) -> R_9(x_9), forall x_9: R_7(x_9) -> R_9(x_9), forall x_9: P_4(x_9) or R_7(x_9) we can infer exists x_9: R_9(x_9) via existential disjunction elimination. Therefore, the answer is yes.