Consider the following premises. For all x_11, if x_11 were a poet, then x_11 is not sleeping. For all x_11, x_11 is rich or x_11 will sleep. There is at least one x_11 for which x_11 will not cook. For all x_11, if x_11 were rich, then x_11 is cooking. Can we infer the following from them? If we can, name the inference rule being used: There is at least one x_11 for which x_11 is not a poet.
 From the fact that for all x_11, if x_11 were rich, then x_11 is cooking, and that there is at least one x_11 for which x_11 will not cook we can infer that there is at least one x_11 for which x_11 is not rich via existential modus tollens. From the fact that for all x_11, x_11 is rich or x_11 will sleep, and that there is at least one x_11 for which x_11 is not rich we can infer that there is at least one x_11 for which x_11 will sleep via existential disjunctive syllogism. Finally, from the fact that for all x_11, if x_11 were a poet, then x_11 is not sleeping, and that there is at least one x_11 for which x_11 will sleep we can infer that there is at least one x_11 for which x_11 is not a poet via existential modus tollens. Therefore, the answer is yes.