Consider the following premises. For all x_10, if x_10 were an astronaut, then x_10 is cooking. For all x_10, if x_10 were bored, then x_10 is cooking. There is at least one x_10 for which x_10 is bored. For all x_10, if x_10 were a poet, then x_10 is not cooking. Can we infer the following from them? If we can, name the inference rule being used: There is at least one x_10 for which x_10 is not a poet.
 From the fact that there is at least one x_10 for which x_10 is bored we can infer that there is at least one x_10 for which x_10 is bored or x_10 is an astronaut via existential addition. From the fact that for all x_10, if x_10 were bored, then x_10 is cooking, that for all x_10, if x_10 were an astronaut, then x_10 is cooking, and that there is at least one x_10 for which x_10 is bored or x_10 is an astronaut we can infer that there is at least one x_10 for which x_10 will cook via existential disjunction elimination. Finally, from the fact that for all x_10, if x_10 were a poet, then x_10 is not cooking, and that there is at least one x_10 for which x_10 will cook we can infer that there is at least one x_10 for which x_10 is not a poet via existential modus tollens. Therefore, the answer is yes.