Consider the following premises. forall x_9: ~P_9(x_9). exists x_9: T(x_9). forall x_9: P_9(x_9) or ~T(x_9). Can we infer exists x_9: Q_5(x_9) from them? If possible, name the inference rules being used at each step.
 From forall x_9: P_9(x_9) or ~T(x_9) we can infer forall x_9: T(x_9) -> P_9(x_9) via universal rewriting. From exists x_9: T(x_9) we can infer exists x_9: T(x_9) or Q_5(x_9) via existential addition. From forall x_9: T(x_9) -> P_9(x_9), forall x_9: ~P_9(x_9) we can infer forall x_9: ~T(x_9) via universal modus tollens. Finally, from exists x_9: T(x_9) or Q_5(x_9), forall x_9: ~T(x_9) we can infer exists x_9: Q_5(x_9) via existential disjunctive syllogism. Therefore, the answer is yes. Notice, however, that the premises were contradictory, so we can infer anything from them.