Consider the following premises. forall x_9: Q_1(x_9) -> P_9(x_9). exists x_9: W_2(x_9) -> Q_1(x_9). forall x_9: P_9(x_9) <-> Q_9(x_9). forall x_9: W_2(x_9) or Q_9(x_9). Can we infer exists x_9: P_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) we can infer forall x_9: Q_9(x_9) -> P_9(x_9) via universal biconditional elimination. From exists x_9: W_2(x_9) -> Q_1(x_9), forall x_9: Q_1(x_9) -> P_9(x_9) we can infer exists x_9: W_2(x_9) -> P_9(x_9) via existential transitivity. Finally, from exists x_9: W_2(x_9) -> P_9(x_9), forall x_9: Q_9(x_9) -> P_9(x_9), forall x_9: W_2(x_9) or Q_9(x_9) we can infer exists x_9: P_9(x_9) via existential disjunction elimination. Therefore, the answer is yes.