Consider the following premises. forall z_2: T(z_2) -> S_1(z_2). forall z_2: S_1(z_2) -> T(z_2). exists z_2: ~S_1(z_2). Can we infer forall z_2: T(z_2) from them? If possible, name the inference rules being used at each step.
 From forall z_2: T(z_2) -> S_1(z_2), forall z_2: S_1(z_2) -> T(z_2) we can infer forall z_2: T(z_2) <-> S_1(z_2) via universal biconditional introduction. Finally, from forall z_2: T(z_2) <-> S_1(z_2), exists z_2: ~S_1(z_2) we can infer exists z_2: ~T(z_2) via existential biconditional elimination, which contradicts forall z_2: T(z_2). Therefore, the answer is no.