Consider the following premises. forall x: P_2(x) -> P_3(x). forall x: P_4(x) -> P_2(x). forall x: P_4(x) <-> P_3(x). Can we infer forall x: P_4(x) <-> P_2(x) from them?
 From forall x: P_4(x) <-> P_3(x) we can infer forall x: P_3(x) -> P_4(x). From forall x: P_2(x) -> P_3(x), forall x: P_3(x) -> P_4(x) we can infer forall x: P_2(x) -> P_4(x). Finally, from forall x: P_4(x) -> P_2(x), forall x: P_2(x) -> P_4(x) we can infer forall x: P_4(x) <-> P_2(x). Therefore, the answer is yes.