Consider the following premises. There is at least one y_2 for which y_2 will drive a car or y_2 is sad. For all y_2, if y_2 drives a car, then y_2 is sleeping. For all y_2, if y_2 sleeps, then y_2 is driving a car. For all y_2, if y_2 were sad, then y_2 is sleeping. Can we infer the following from them? If we can, name the inference rule being used: For all y_2, y_2 will not sleep.
 From the fact that for all y_2, if y_2 drives a car, then y_2 is sleeping, and that for all y_2, if y_2 sleeps, then y_2 is driving a car we can infer that for all y_2, y_2 will drive a car if and only if y_2 will sleep via universal biconditional introduction. From the fact that for all y_2, y_2 will drive a car if and only if y_2 will sleep we can infer that for all y_2, if y_2 drives a car, then y_2 is sleeping via universal biconditional elimination. Finally, from the fact that for all y_2, if y_2 drives a car, then y_2 is sleeping, that for all y_2, if y_2 were sad, then y_2 is sleeping, and that there is at least one y_2 for which y_2 will drive a car or y_2 is sad we can infer that there is at least one y_2 for which y_2 will sleep via existential disjunction elimination, which contradicts that For all y_2, y_2 will not sleep. Therefore, the answer is no.