Consider the following premises. forall x_9: P_4(x_9) -> Q_4(x_9). forall x_9: ~P_9(x_9). forall x_9: P_4(x_9). forall x_9: P_9(x_9) or P_1(x_9). Can we infer P_1(e_1) from them? If possible, name the inference rules being used at each step.
 From forall x_9: P_9(x_9) or P_1(x_9), forall x_9: ~P_9(x_9) we can infer forall x_9: P_1(x_9) via universal disjunctive syllogism. From forall x_9: P_4(x_9) -> Q_4(x_9), forall x_9: P_4(x_9) we can infer forall x_9: Q_4(x_9) via universal modus ponens. From forall x_9: P_1(x_9), forall x_9: Q_4(x_9) we can infer forall x_9: P_1(x_9) and Q_4(x_9) via universal conjunction. From forall x_9: P_1(x_9) and Q_4(x_9) we can infer P_1(e_1) and Q_4(e_1) via universal instantiation. Finally, from P_1(e_1) and Q_4(e_1) we can infer P_1(e_1) via simplification. Therefore, the answer is yes.