Consider the following premises. forall x_9: P_9(x_9) -> Q_9(x_9). forall x_9: Q_9(x_9) -> R_9(x_9). forall x_9: R_9(x_9) -> P_9(x_9). R_9(a_7) -> P_9(a_7). Can we infer P_9(a_7) <-> R_9(a_7) 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), forall x_9: Q_9(x_9) -> R_9(x_9) we can infer forall x_9: P_9(x_9) -> R_9(x_9) via universal transitivity. From forall x_9: R_9(x_9) -> P_9(x_9), forall x_9: P_9(x_9) -> R_9(x_9) we can infer forall x_9: R_9(x_9) <-> P_9(x_9) via universal biconditional introduction. From forall x_9: R_9(x_9) <-> P_9(x_9) we can infer R_9(a_7) <-> P_9(a_7) via universal instantiation. From R_9(a_7) <-> P_9(a_7) we can infer P_9(a_7) -> R_9(a_7) via biconditional elimination. Finally, from P_9(a_7) -> R_9(a_7), R_9(a_7) -> P_9(a_7) we can infer P_9(a_7) <-> R_9(a_7) via biconditional introduction. Therefore, the answer is yes.