Consider the following premises. exists z: T(z) <-> S(z). forall z: T(z) or ~T_4(z). forall z: S(z) -> T_4(z). Can we infer forall z: ~(S(z) <-> T(z)) from them? If possible, name the inference rules being used at each step.
 From forall z: T(z) or ~T_4(z) we can infer forall z: T_4(z) -> T(z) via universal rewriting. From forall z: S(z) -> T_4(z), forall z: T_4(z) -> T(z) we can infer forall z: S(z) -> T(z) via universal transitivity. From exists z: T(z) <-> S(z) we can infer exists z: T(z) -> S(z) via existential biconditional elimination. Finally, from forall z: S(z) -> T(z), exists z: T(z) -> S(z) we can infer exists z: S(z) <-> T(z) via existential biconditional introduction, which contradicts forall z: ~(S(z) <-> T(z)). Therefore, the answer is no.