Consider the following premises. forall x_10: Q_1(x_10) -> P_4(x_10). forall x_10: Q(x_10) -> ~P_4(x_10). exists x_10: U_3(x_10). forall x_10: U_3(x_10) -> P_4(x_10). Can we infer exists x_10: ~Q(x_10) from them? If possible, name the inference rules being used at each step.
 From exists x_10: U_3(x_10) we can infer exists x_10: U_3(x_10) or Q_1(x_10) via existential addition. From forall x_10: U_3(x_10) -> P_4(x_10), forall x_10: Q_1(x_10) -> P_4(x_10), exists x_10: U_3(x_10) or Q_1(x_10) we can infer exists x_10: P_4(x_10) via existential disjunction elimination. Finally, from forall x_10: Q(x_10) -> ~P_4(x_10), exists x_10: P_4(x_10) we can infer exists x_10: ~Q(x_10) via existential modus tollens. Therefore, the answer is yes.