Consider the following premises. forall x: W_2(x) or Q(x). forall x: W_2(x) -> Q_1(x). forall x: P_4(x). forall x: ~Q_1(x). Can we infer exists x: ~(P_4(x) and Q(x)) from them? If possible, name the inference rules being used at each step.
 From forall x: W_2(x) -> Q_1(x), forall x: ~Q_1(x) we can infer forall x: ~W_2(x) via universal modus tollens. From forall x: W_2(x) or Q(x), forall x: ~W_2(x) we can infer forall x: Q(x) via universal disjunctive syllogism. Finally, from forall x: P_4(x), forall x: Q(x) we can infer forall x: P_4(x) and Q(x) via universal conjunction, which contradicts exists x: ~(P_4(x) and Q(x)). Therefore, the answer is no.