Consider the following premises. forall x: U(x) or Q_4(x). forall x: Q_4(x) <-> R(x). forall x: U(x) -> S_2(x). forall x: R(x) or ~S_2(x). Can we infer exists x: ~R(x) from them? If possible, name the inference rules being used at each step.
 From forall x: R(x) or ~S_2(x) we can infer forall x: S_2(x) -> R(x) via universal rewriting. From forall x: Q_4(x) <-> R(x) we can infer forall x: Q_4(x) -> R(x) via universal biconditional elimination. From forall x: U(x) -> S_2(x), forall x: S_2(x) -> R(x) we can infer forall x: U(x) -> R(x) via universal transitivity. Finally, from forall x: U(x) -> R(x), forall x: Q_4(x) -> R(x), forall x: U(x) or Q_4(x) we can infer forall x: R(x) via universal disjunction elimination, which contradicts exists x: ~R(x). Therefore, the answer is no.