Consider the following premises. forall x_8: T(x_8). exists x_8: ~T(x_8) or R_1(x_8). forall x_8: P(x_8) -> R(x_8). forall x_8: R(x_8) or ~R_1(x_8). Can we infer forall x_8: ~R(x_8) from them? If possible, name the inference rules being used at each step.
 From forall x_8: T(x_8) we can infer forall x_8: T(x_8) or P(x_8) via universal addition. From forall x_8: T(x_8) or P(x_8), exists x_8: ~T(x_8) or R_1(x_8) we can infer exists x_8: P(x_8) or R_1(x_8) via existential resolution. From forall x_8: R(x_8) or ~R_1(x_8) we can infer forall x_8: R_1(x_8) -> R(x_8) via universal rewriting. Finally, from forall x_8: P(x_8) -> R(x_8), forall x_8: R_1(x_8) -> R(x_8), exists x_8: P(x_8) or R_1(x_8) we can infer exists x_8: R(x_8) via existential disjunction elimination, which contradicts forall x_8: ~R(x_8). Therefore, the answer is no.