Consider the following premises. forall x_4: R_4(x_4) -> P_4(x_4). forall x_4: P_4(x_4) -> Q_4(x_4). forall x_4: Q_4(x_4) -> R_4(x_4). Can we infer P_4(a_2) -> R_4(a_2) from them? If possible, name the inference rules being used at each step.
 From forall x_4: P_4(x_4) -> Q_4(x_4), forall x_4: Q_4(x_4) -> R_4(x_4) we can infer forall x_4: P_4(x_4) -> R_4(x_4) via universal transitivity. From forall x_4: R_4(x_4) -> P_4(x_4), forall x_4: P_4(x_4) -> R_4(x_4) we can infer forall x_4: R_4(x_4) <-> P_4(x_4) via universal biconditional introduction. From forall x_4: R_4(x_4) <-> P_4(x_4) we can infer R_4(a_2) <-> P_4(a_2) via universal instantiation. Finally, from R_4(a_2) <-> P_4(a_2) we can infer P_4(a_2) -> R_4(a_2) via biconditional elimination. Therefore, the answer is yes.