Consider the following premises. forall x_18: P_18(x_18) or ~Q_18(x_18). forall x_18: P_2(x_18) or Q_2(x_18). forall x_18: R_15(x_18) -> P_18(x_18). forall x_18: ~P_15(x_18) or R_15(x_18). forall x_18: ~P_2(x_18). exists x_18: P_15(x_18) or Q_18(x_18). Can we infer forall x_18: ~(Q_2(x_18) and P_18(x_18)) from them? If possible, name the inference rules being used at each step.
 From forall x_18: P_18(x_18) or ~Q_18(x_18) we can infer forall x_18: Q_18(x_18) -> P_18(x_18) via universal rewriting. From exists x_18: P_15(x_18) or Q_18(x_18), forall x_18: ~P_15(x_18) or R_15(x_18) we can infer exists x_18: Q_18(x_18) or R_15(x_18) via existential resolution. From forall x_18: Q_18(x_18) -> P_18(x_18), forall x_18: R_15(x_18) -> P_18(x_18), exists x_18: Q_18(x_18) or R_15(x_18) we can infer exists x_18: P_18(x_18) via existential disjunction elimination. From forall x_18: P_2(x_18) or Q_2(x_18), forall x_18: ~P_2(x_18) we can infer forall x_18: Q_2(x_18) via universal disjunctive syllogism. Finally, from forall x_18: Q_2(x_18), exists x_18: P_18(x_18) we can infer exists x_18: Q_2(x_18) and P_18(x_18) via existential conjunction, which contradicts forall x_18: ~(Q_2(x_18) and P_18(x_18)). Therefore, the answer is no.