Consider the following premises. forall x: P_1(x) -> T(x). exists x: T(x) -> R(x). forall x: R(x) <-> P_2(x). Can we infer exists x: P_1(x) -> P_2(x) from them? If possible, name the inference rules being used at each step.
 From forall x: P_1(x) -> T(x), exists x: T(x) -> R(x) we can infer exists x: P_1(x) -> R(x) via existential transitivity. From forall x: R(x) <-> P_2(x) we can infer forall x: R(x) -> P_2(x) via universal biconditional elimination. Finally, from exists x: P_1(x) -> R(x), forall x: R(x) -> P_2(x) we can infer exists x: P_1(x) -> P_2(x) via existential transitivity. Therefore, the answer is yes.