Consider the following premises. forall x: P(x) -> R(x). forall x: P_2(x) -> P(x). forall x: R(x) -> P_2(x). Can we infer forall x: P(x) <-> P_2(x) from them? If possible, name the inference rules being used at each step.
 From forall x: P(x) -> R(x), forall x: R(x) -> P_2(x) we can infer forall x: P(x) -> P_2(x) via universal transitivity. Finally, from forall x: P(x) -> P_2(x), forall x: P_2(x) -> P(x) we can infer forall x: P(x) <-> P_2(x) via universal biconditional introduction. Therefore, the answer is yes.