Consider the following premises. exists z: U_4(z) or ~U(z). forall z: U(z). forall z: U_4(z) -> S(z). Can we infer exists z: S(z) from them? If possible, name the inference rules being used at each step.
 From exists z: U_4(z) or ~U(z) we can infer exists z: U(z) -> U_4(z) via existential rewriting. From exists z: U(z) -> U_4(z), forall z: U_4(z) -> S(z) we can infer exists z: U(z) -> S(z) via existential transitivity. Finally, from exists z: U(z) -> S(z), forall z: U(z) we can infer exists z: S(z) via existential modus ponens. Therefore, the answer is yes.