Consider the following premises. forall x: P(x) -> R_3(x). forall x: R_3(x) -> T(x). forall x: T(x) -> P(x). forall x: T(x) -> R_3(x). Can we infer forall x: T(x) <-> P(x) from them? If possible, name the inference rules being used at each step.
 From forall x: R_3(x) -> T(x), forall x: T(x) -> R_3(x) we can infer forall x: R_3(x) <-> T(x) via universal biconditional introduction. From forall x: R_3(x) <-> T(x) we can infer forall x: R_3(x) -> T(x) via universal biconditional elimination. From forall x: P(x) -> R_3(x), forall x: R_3(x) -> T(x) we can infer forall x: P(x) -> T(x) via universal transitivity. Finally, from forall x: T(x) -> P(x), forall x: P(x) -> T(x) we can infer forall x: T(x) <-> P(x) via universal biconditional introduction. Therefore, the answer is yes.