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