Consider the following premises. forall x: U_4(x) or ~S_4(x). forall x: S(x) or U(x). forall x: S(x) -> S_4(x). exists x: U(x) -> S_4(x). Can we infer exists x: U_4(x) from them? If possible, name the inference rules being used at each step.
 From forall x: S(x) -> S_4(x), exists x: U(x) -> S_4(x), forall x: S(x) or U(x) we can infer exists x: S_4(x) via existential disjunction elimination. From forall x: U_4(x) or ~S_4(x) we can infer forall x: S_4(x) -> U_4(x) via universal rewriting. Finally, from forall x: S_4(x) -> U_4(x), exists x: S_4(x) we can infer exists x: U_4(x) via existential modus ponens. Therefore, the answer is yes.