Consider the following premises. exists x_12: R_11(x_12) -> R_8(x_12). forall x_12: Q_11(x_12) -> R_8(x_12). forall x_12: ~P_11(x_12) or R_11(x_12). forall x_12: P_11(x_12) or Q_11(x_12). forall x_12: P_12(x_12) -> Q_4(x_12). forall x_12: P_12(x_12). Can we infer forall x_12: ~(Q_4(x_12) and R_8(x_12)) from them? If possible, name the inference rules being used at each step.
 From forall x_12: P_12(x_12) -> Q_4(x_12), forall x_12: P_12(x_12) we can infer forall x_12: Q_4(x_12) via universal modus ponens. From forall x_12: P_11(x_12) or Q_11(x_12), forall x_12: ~P_11(x_12) or R_11(x_12) we can infer forall x_12: Q_11(x_12) or R_11(x_12) via universal resolution. From forall x_12: Q_11(x_12) -> R_8(x_12), exists x_12: R_11(x_12) -> R_8(x_12), forall x_12: Q_11(x_12) or R_11(x_12) we can infer exists x_12: R_8(x_12) via existential disjunction elimination. Finally, from forall x_12: Q_4(x_12), exists x_12: R_8(x_12) we can infer exists x_12: Q_4(x_12) and R_8(x_12) via existential conjunction, which contradicts forall x_12: ~(Q_4(x_12) and R_8(x_12)). Therefore, the answer is no.