Consider the following premises. exists x: T(x) -> T_3(x). forall x: P(x) <-> T_3(x). forall x: P(x) -> T(x). Can we infer exists x: T_3(x) -> P(x) from them? If possible, name the inference rules being used at each step.
 From forall x: P(x) -> T(x), exists x: T(x) -> T_3(x) we can infer exists x: P(x) -> T_3(x) via existential transitivity. From forall x: P(x) <-> T_3(x) we can infer forall x: T_3(x) -> P(x) via universal biconditional elimination. From forall x: T_3(x) -> P(x), exists x: P(x) -> T_3(x) we can infer exists x: T_3(x) <-> P(x) via existential biconditional introduction. Finally, from exists x: T_3(x) <-> P(x) we can infer exists x: T_3(x) -> P(x) via existential biconditional elimination. Therefore, the answer is yes.