Consider the following premises. forall z: S_4(z) <-> Q_1(z). forall z: T(z) or ~S_4(z). exists z: T(z) -> Q_1(z). Can we infer exists z: T(z) <-> S_4(z) from them? If possible, name the inference rules being used at each step.
 From forall z: T(z) or ~S_4(z) we can infer forall z: S_4(z) -> T(z) via universal rewriting. From forall z: S_4(z) <-> Q_1(z) we can infer forall z: Q_1(z) -> S_4(z) via universal biconditional elimination. From exists z: T(z) -> Q_1(z), forall z: Q_1(z) -> S_4(z) we can infer exists z: T(z) -> S_4(z) via existential transitivity. Finally, from exists z: T(z) -> S_4(z), forall z: S_4(z) -> T(z) we can infer exists z: T(z) <-> S_4(z) via existential biconditional introduction. Therefore, the answer is yes.