Consider the following premises. exists x: T_4(x) or ~S_2(x). forall x: Q(x) or S_2(x). forall x: ~Q(x) or W(x). forall x: T_4(x) -> R_4(x). forall x: R_4(x) or ~W(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: Q(x) or S_2(x), forall x: ~Q(x) or W(x) we can infer forall x: S_2(x) or W(x) via universal resolution. From exists x: T_4(x) or ~S_2(x) we can infer exists x: S_2(x) -> T_4(x) via existential rewriting. From exists x: S_2(x) -> T_4(x), forall x: T_4(x) -> R_4(x) we can infer exists x: S_2(x) -> R_4(x) via existential transitivity. 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: S_2(x) -> R_4(x), forall x: W(x) -> R_4(x), forall x: S_2(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.