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