Consider the following premises. exists x: U(x) -> R_4(x). forall x: R_4(x) or ~W(x). forall x: U(x). Can we infer forall x: ~R_4(x) from them? If possible, name the inference rules being used at each step.
 From forall x: U(x) we can infer forall x: U(x) or W(x) via universal addition. From forall x: R_4(x) or ~W(x) we can infer forall x: W(x) -> R_4(x) via universal rewriting. Finally, from exists x: U(x) -> R_4(x), forall x: W(x) -> R_4(x), forall x: U(x) or W(x) we can infer exists x: R_4(x) via existential disjunction elimination, which contradicts forall x: ~R_4(x). Therefore, the answer is no.