Consider the following premises. exists x_13: Q_9(x_13) -> U_1(x_13). forall x_13: Q_13(x_13) -> Q(x_13). forall x_13: U_1(x_13) -> Q_13(x_13). forall x_13: ~Q(x_13). forall x_13: Q(x_13) -> Q_9(x_13). Can we infer exists x_13: ~U_1(x_13) from them? If possible, name the inference rules being used at each step.
 From forall x_13: U_1(x_13) -> Q_13(x_13), forall x_13: Q_13(x_13) -> Q(x_13) we can infer forall x_13: U_1(x_13) -> Q(x_13) via universal transitivity. From forall x_13: Q(x_13) -> Q_9(x_13), exists x_13: Q_9(x_13) -> U_1(x_13) we can infer exists x_13: Q(x_13) -> U_1(x_13) via existential transitivity. From forall x_13: U_1(x_13) -> Q(x_13), exists x_13: Q(x_13) -> U_1(x_13) we can infer exists x_13: U_1(x_13) <-> Q(x_13) via existential biconditional introduction. From exists x_13: U_1(x_13) <-> Q(x_13) we can infer exists x_13: U_1(x_13) -> Q(x_13) via existential biconditional elimination. Finally, from exists x_13: U_1(x_13) -> Q(x_13), forall x_13: ~Q(x_13) we can infer exists x_13: ~U_1(x_13) via existential modus tollens. Therefore, the answer is yes.