Consider the following premises. exists x_12: Q_12(x_12) -> Q(x_12). forall x_12: S_2(x_12) -> T(x_12). forall x_12: R(x_12) or Q_12(x_12). forall x_12: S_2(x_12). forall x_12: R(x_12) -> Q(x_12). Can we infer forall x_12: ~(T(x_12) and Q(x_12)) from them? If possible, name the inference rules being used at each step.
 From forall x_12: R(x_12) -> Q(x_12), exists x_12: Q_12(x_12) -> Q(x_12), forall x_12: R(x_12) or Q_12(x_12) we can infer exists x_12: Q(x_12) via existential disjunction elimination. From forall x_12: S_2(x_12) -> T(x_12), forall x_12: S_2(x_12) we can infer forall x_12: T(x_12) via universal modus ponens. Finally, from forall x_12: T(x_12), exists x_12: Q(x_12) we can infer exists x_12: T(x_12) and Q(x_12) via existential conjunction, which contradicts forall x_12: ~(T(x_12) and Q(x_12)). Therefore, the answer is no.