Consider the following premises. exists z: ~R(z). forall z: R(z) -> S_4(z). forall z: S_4(z) <-> R(z). Can we infer exists z: ~S_4(z) from them?
 From forall z: S_4(z) <-> R(z) we can infer forall z: S_4(z) -> R(z). From forall z: R(z) -> S_4(z), forall z: S_4(z) -> R(z) we can infer forall z: R(z) <-> S_4(z). Finally, from forall z: R(z) <-> S_4(z), exists z: ~R(z) we can infer exists z: ~S_4(z). Therefore, the answer is yes.