Consider the following premises. exists x_9: Q_2(x_9) -> S(x_9). forall x_9: P_9(x_9) <-> S(x_9). forall x_9: P_9(x_9) -> Q_2(x_9). Can we infer exists x_9: S(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) <-> S(x_9) we can infer forall x_9: S(x_9) -> P_9(x_9) via universal biconditional elimination. From forall x_9: P_9(x_9) -> Q_2(x_9), exists x_9: Q_2(x_9) -> S(x_9) we can infer exists x_9: P_9(x_9) -> S(x_9) via existential transitivity. Finally, from forall x_9: S(x_9) -> P_9(x_9), exists x_9: P_9(x_9) -> S(x_9) we can infer exists x_9: S(x_9) <-> P_9(x_9) via existential biconditional introduction. Therefore, the answer is yes.