Consider the following premises. exists x_5: W_2(x_5). forall x_5: ~Q_5(x_5). forall x_5: P_5(x_5) -> Q_5(x_5). forall x_5: P_5(x_5) or Q(x_5). Can we infer forall x_5: ~(W_2(x_5) and Q(x_5)) from them? If possible, name the inference rules being used at each step.
 From forall x_5: P_5(x_5) -> Q_5(x_5), forall x_5: ~Q_5(x_5) we can infer forall x_5: ~P_5(x_5) via universal modus tollens. From forall x_5: P_5(x_5) or Q(x_5), forall x_5: ~P_5(x_5) we can infer forall x_5: Q(x_5) via universal disjunctive syllogism. Finally, from exists x_5: W_2(x_5), forall x_5: Q(x_5) we can infer exists x_5: W_2(x_5) and Q(x_5) via existential conjunction, which contradicts forall x_5: ~(W_2(x_5) and Q(x_5)). Therefore, the answer is no.