Consider the following premises. forall x_6: Q(x_6). forall x_6: W_2(x_6) or ~Q_6(x_6). forall x_6: Q_6(x_6) or Q_4(x_6). forall x_6: Q_4(x_6) -> W_2(x_6). Can we infer exists x_6: ~(Q(x_6) and W_2(x_6)) from them? If possible, name the inference rules being used at each step.
 From forall x_6: W_2(x_6) or ~Q_6(x_6) we can infer forall x_6: Q_6(x_6) -> W_2(x_6) via universal rewriting. From forall x_6: Q_6(x_6) -> W_2(x_6), forall x_6: Q_4(x_6) -> W_2(x_6), forall x_6: Q_6(x_6) or Q_4(x_6) we can infer forall x_6: W_2(x_6) via universal disjunction elimination. Finally, from forall x_6: Q(x_6), forall x_6: W_2(x_6) we can infer forall x_6: Q(x_6) and W_2(x_6) via universal conjunction, which contradicts exists x_6: ~(Q(x_6) and W_2(x_6)). Therefore, the answer is no.