Consider the following premises. exists x: T(x) -> T_4(x). forall x: T_4(x) -> T(x). forall x: T(x) -> W_1(x). Can we infer exists x: T_4(x) -> W_1(x) from them? If possible, name the inference rules being used at each step.
 From forall x: T_4(x) -> T(x), exists x: T(x) -> T_4(x) we can infer exists x: T_4(x) <-> T(x) via existential biconditional introduction. From exists x: T_4(x) <-> T(x) we can infer exists x: T_4(x) -> T(x) via existential biconditional elimination. Finally, from exists x: T_4(x) -> T(x), forall x: T(x) -> W_1(x) we can infer exists x: T_4(x) -> W_1(x) via existential transitivity. Therefore, the answer is yes.