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