Consider the following premises. exists x_12: Q_7(x_12) -> Q_12(x_12). forall x_12: T(x_12) or ~Q_12(x_12). forall x_12: ~T(x_12). forall x_12: U(x_12) -> Q_7(x_12). Can we infer exists x_12: ~U(x_12) from them? If possible, name the inference rules being used at each step.
 From forall x_12: T(x_12) or ~Q_12(x_12) we can infer forall x_12: Q_12(x_12) -> T(x_12) via universal rewriting. From forall x_12: U(x_12) -> Q_7(x_12), exists x_12: Q_7(x_12) -> Q_12(x_12) we can infer exists x_12: U(x_12) -> Q_12(x_12) via existential transitivity. From exists x_12: U(x_12) -> Q_12(x_12), forall x_12: Q_12(x_12) -> T(x_12) we can infer exists x_12: U(x_12) -> T(x_12) via existential transitivity. Finally, from exists x_12: U(x_12) -> T(x_12), forall x_12: ~T(x_12) we can infer exists x_12: ~U(x_12) via existential modus tollens. Therefore, the answer is yes.