Consider the following premises. forall x_12: T(x_12) -> Q_7(x_12). forall x_12: ~S(x_12). forall x_12: S(x_12) or ~Q(x_12). exists x_12: Q_7(x_12) -> Q(x_12). Can we infer forall x_12: T(x_12) from them? If possible, name the inference rules being used at each step.
 From forall x_12: S(x_12) or ~Q(x_12) we can infer forall x_12: Q(x_12) -> S(x_12) via universal rewriting. From forall x_12: T(x_12) -> Q_7(x_12), exists x_12: Q_7(x_12) -> Q(x_12) we can infer exists x_12: T(x_12) -> Q(x_12) via existential transitivity. From exists x_12: T(x_12) -> Q(x_12), forall x_12: Q(x_12) -> S(x_12) we can infer exists x_12: T(x_12) -> S(x_12) via existential transitivity. Finally, from exists x_12: T(x_12) -> S(x_12), forall x_12: ~S(x_12) we can infer exists x_12: ~T(x_12) via existential modus tollens, which contradicts forall x_12: T(x_12). Therefore, the answer is no.