Consider the following premises. exists x_9: P_5(x_9) <-> W_4(x_9). forall x_9: P_9(x_9) or Q_9(x_9). forall x_9: Q_9(x_9) -> W_4(x_9). forall x_9: P_9(x_9) -> W_4(x_9). Can we infer exists x_9: P_5(x_9) from them? If possible, name the inference rules being used at each step.
 From forall x_9: P_9(x_9) -> W_4(x_9), forall x_9: Q_9(x_9) -> W_4(x_9), forall x_9: P_9(x_9) or Q_9(x_9) we can infer forall x_9: W_4(x_9) via universal disjunction elimination. From exists x_9: P_5(x_9) <-> W_4(x_9) we can infer exists x_9: W_4(x_9) -> P_5(x_9) via existential biconditional elimination. Finally, from exists x_9: W_4(x_9) -> P_5(x_9), forall x_9: W_4(x_9) we can infer exists x_9: P_5(x_9) via existential modus ponens. Therefore, the answer is yes.