Consider the following premises. forall x_9: ~W_1(x_9). forall x_9: Q(x_9) -> Q_2(x_9). forall x_9: Q_2(x_9) <-> W_1(x_9). exists x_9: Q_2(x_9) -> Q(x_9). Can we infer forall x_9: Q(x_9) from them? If possible, name the inference rules being used at each step.
 From forall x_9: Q_2(x_9) <-> W_1(x_9) we can infer forall x_9: Q_2(x_9) -> W_1(x_9) via universal biconditional elimination. From forall x_9: Q(x_9) -> Q_2(x_9), exists x_9: Q_2(x_9) -> Q(x_9) we can infer exists x_9: Q(x_9) <-> Q_2(x_9) via existential biconditional introduction. From forall x_9: Q_2(x_9) -> W_1(x_9), forall x_9: ~W_1(x_9) we can infer forall x_9: ~Q_2(x_9) via universal modus tollens. Finally, from exists x_9: Q(x_9) <-> Q_2(x_9), forall x_9: ~Q_2(x_9) we can infer exists x_9: ~Q(x_9) via existential biconditional elimination, which contradicts forall x_9: Q(x_9). Therefore, the answer is no.