Consider the following premises. forall x_10: ~P_10(x_10). forall x_10: P_1(x_10) -> Q_10(x_10). forall x_10: W_3(x_10) -> P_1(x_10). forall x_10: P_10(x_10) or ~Q_10(x_10). Can we infer exists x_10: W_3(x_10) from them? If possible, name the inference rules being used at each step.
 From forall x_10: P_10(x_10) or ~Q_10(x_10) we can infer forall x_10: Q_10(x_10) -> P_10(x_10) via universal rewriting. From forall x_10: Q_10(x_10) -> P_10(x_10), forall x_10: ~P_10(x_10) we can infer forall x_10: ~Q_10(x_10) via universal modus tollens. From forall x_10: P_1(x_10) -> Q_10(x_10), forall x_10: ~Q_10(x_10) we can infer forall x_10: ~P_1(x_10) via universal modus tollens. Finally, from forall x_10: W_3(x_10) -> P_1(x_10), forall x_10: ~P_1(x_10) we can infer forall x_10: ~W_3(x_10) via universal modus tollens, which contradicts exists x_10: W_3(x_10). Therefore, the answer is no.