Consider the following premises. exists x_8: ~P_8(x_8) or T(x_8). forall x_8: P_8(x_8). forall x_8: P_2(x_8) or ~T(x_8). forall x_8: R_2(x_8) -> P_2(x_8). Can we infer forall x_8: ~P_2(x_8) from them? If possible, name the inference rules being used at each step.
 From forall x_8: P_8(x_8) we can infer forall x_8: P_8(x_8) or R_2(x_8) via universal addition. From forall x_8: P_8(x_8) or R_2(x_8), exists x_8: ~P_8(x_8) or T(x_8) we can infer exists x_8: R_2(x_8) or T(x_8) via existential resolution. From forall x_8: P_2(x_8) or ~T(x_8) we can infer forall x_8: T(x_8) -> P_2(x_8) via universal rewriting. Finally, from forall x_8: R_2(x_8) -> P_2(x_8), forall x_8: T(x_8) -> P_2(x_8), exists x_8: R_2(x_8) or T(x_8) we can infer exists x_8: P_2(x_8) via existential disjunction elimination, which contradicts forall x_8: ~P_2(x_8). Therefore, the answer is no.