Consider the following premises. For all x_10, x_10 is not an astronaut or x_10 will sleep. For all x_10, if x_10 were a bartender, then x_10 is not an office clerk. There is at least one x_10 for which x_10 is an office clerk. For all x_10, x_10 is an astronaut or x_10 is a bartender. 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 will sleep.
 From the fact that for all x_10, x_10 is an astronaut or x_10 is a bartender, and that for all x_10, x_10 is not an astronaut or x_10 will sleep we can infer that for all x_10, x_10 is a bartender or x_10 will sleep via universal resolution. From the fact that for all x_10, if x_10 were a bartender, then x_10 is not an office clerk, and that there is at least one x_10 for which x_10 is an office clerk we can infer that there is at least one x_10 for which x_10 is not a bartender via existential modus tollens. Finally, from the fact that for all x_10, x_10 is a bartender or x_10 will sleep, and that there is at least one x_10 for which x_10 is not a bartender we can infer that there is at least one x_10 for which x_10 will sleep via existential disjunctive syllogism. Therefore, the answer is yes.