Consider the following premises. forall z: S_3(z) -> T_2(z). exists z: S(z) or ~S_3(z). forall z: T_2(z) -> S_3(z). forall z: T(z) -> T_2(z). Can we infer exists z: T(z) -> S(z) from them? If possible, name the inference rules being used at each step.
 From forall z: S_3(z) -> T_2(z), forall z: T_2(z) -> S_3(z) we can infer forall z: S_3(z) <-> T_2(z) via universal biconditional introduction. From exists z: S(z) or ~S_3(z) we can infer exists z: S_3(z) -> S(z) via existential rewriting. From forall z: S_3(z) <-> T_2(z) we can infer forall z: T_2(z) -> S_3(z) via universal biconditional elimination. From forall z: T(z) -> T_2(z), forall z: T_2(z) -> S_3(z) we can infer forall z: T(z) -> S_3(z) via universal transitivity. Finally, from forall z: T(z) -> S_3(z), exists z: S_3(z) -> S(z) we can infer exists z: T(z) -> S(z) via existential transitivity. Therefore, the answer is yes.