Consider the following premises. For all x_11, x_11 is an astronaut or x_11 is a nurse. There is at least one x_11 for which if x_11 were an astronaut, then x_11 is working. For all x_11, x_11 will cook or x_11 is a janitor. For all x_11, if x_11 were a nurse, then x_11 is working. For all x_11, x_11 will not cook. 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 will work and x_11 is a janitor.
 From the fact that there is at least one x_11 for which if x_11 were an astronaut, then x_11 is working, that for all x_11, if x_11 were a nurse, then x_11 is working, and that for all x_11, x_11 is an astronaut or x_11 is a nurse we can infer that there is at least one x_11 for which x_11 will work via existential disjunction elimination. From the fact that for all x_11, x_11 will cook or x_11 is a janitor, and that for all x_11, x_11 will not cook we can infer that for all x_11, x_11 is a janitor via universal disjunctive syllogism. Finally, from the fact that there is at least one x_11 for which x_11 will work, and that for all x_11, x_11 is a janitor we can infer that there is at least one x_11 for which x_11 will work and x_11 is a janitor via existential conjunction. Therefore, the answer is yes.