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