Consider the following premises. For all y, if y climbs a mountain, then y is sleeping. There is at least one y for which y will drive a car if and only if y will sleep. For all y, y will drive a car or y is a scientist. For all y, if y were a scientist, then y is climbing a mountain. Can we infer the following from them? If we can, name the inference rule being used: There is at least one y for which y will sleep.
 From the fact that for all y, if y were a scientist, then y is climbing a mountain, and that for all y, if y climbs a mountain, then y is sleeping we can infer that for all y, if y were a scientist, then y is sleeping via universal transitivity. From the fact that there is at least one y for which y will drive a car if and only if y will sleep we can infer that there is at least one y for which if y drives a car, then y is sleeping via existential biconditional elimination. Finally, from the fact that there is at least one y for which if y drives a car, then y is sleeping, that for all y, if y were a scientist, then y is sleeping, and that for all y, y will drive a car or y is a scientist we can infer that there is at least one y for which y will sleep via existential disjunction elimination. Therefore, the answer is yes.