Consider the following premises. forall x_9: ~P_9(x_9). exists x_9: P_7(x_9) <-> W(x_9). forall x_9: Q_1(x_9) -> P_7(x_9). forall x_9: P_9(x_9) or W(x_9). Can we infer exists x_9: P_7(x_9) from them? If possible, name the inference rules being used at each step.
 From forall x_9: P_9(x_9) or W(x_9), forall x_9: ~P_9(x_9) we can infer forall x_9: W(x_9) via universal disjunctive syllogism. From exists x_9: P_7(x_9) <-> W(x_9) we can infer exists x_9: W(x_9) -> P_7(x_9) via existential biconditional elimination. From forall x_9: W(x_9) we can infer forall x_9: W(x_9) or Q_1(x_9) via universal addition. Finally, from exists x_9: W(x_9) -> P_7(x_9), forall x_9: Q_1(x_9) -> P_7(x_9), forall x_9: W(x_9) or Q_1(x_9) we can infer exists x_9: P_7(x_9) via existential disjunction elimination. Therefore, the answer is yes.