Consider the following premises. exists z_2: R(z_2) or T(z_2). forall z_2: T(z_2) -> Q(z_2). forall z_2: R(z_2) <-> S_4(z_2). forall z_2: Q(z_2) -> S_4(z_2). Can we infer forall z_2: ~S_4(z_2) from them? If possible, name the inference rules being used at each step.
 From forall z_2: R(z_2) <-> S_4(z_2) we can infer forall z_2: R(z_2) -> S_4(z_2) via universal biconditional elimination. From forall z_2: T(z_2) -> Q(z_2), forall z_2: Q(z_2) -> S_4(z_2) we can infer forall z_2: T(z_2) -> S_4(z_2) via universal transitivity. Finally, from forall z_2: R(z_2) -> S_4(z_2), forall z_2: T(z_2) -> S_4(z_2), exists z_2: R(z_2) or T(z_2) we can infer exists z_2: S_4(z_2) via existential disjunction elimination, which contradicts forall z_2: ~S_4(z_2). Therefore, the answer is no.