Consider the following premises. forall z: T(z) -> S_4(z). forall z: T(z) -> S_4(z). exists z: S_4(z) -> T(z). Can we infer exists z: S_4(z) <-> T(z) from them? If possible, name the inference rules being used at each step.
 From forall z: T(z) -> S_4(z), exists z: S_4(z) -> T(z) we can infer exists z: T(z) <-> S_4(z) via existential biconditional introduction. From exists z: T(z) <-> S_4(z) we can infer exists z: S_4(z) -> T(z) via existential biconditional elimination. Finally, from exists z: S_4(z) -> T(z), forall z: T(z) -> S_4(z) we can infer exists z: S_4(z) <-> T(z) via existential biconditional introduction. Therefore, the answer is yes.