Consider the following premises. For all x_12, x_12 will sleep. For all x_12, x_12 is poor or x_12 is bored. There is at least one x_12 for which if x_12 were poor, then x_12 is not sleeping. Can we infer the following from them? If we can, name the inference rule being used: There is at least one x_12 for which x_12 is bored or x_12 will cook.
 From the fact that there is at least one x_12 for which if x_12 were poor, then x_12 is not sleeping, and that for all x_12, x_12 will sleep we can infer that there is at least one x_12 for which x_12 is not poor via existential modus tollens. From the fact that for all x_12, x_12 is poor or x_12 is bored, and that there is at least one x_12 for which x_12 is not poor we can infer that there is at least one x_12 for which x_12 is bored via existential disjunctive syllogism. Finally, from the fact that there is at least one x_12 for which x_12 is bored we can infer that there is at least one x_12 for which x_12 is bored or x_12 will cook via existential addition. Therefore, the answer is yes.