Consider the following premises. There is at least one y for which if y climbs a mountain, then y is drinking water. For all y, y will listen to a song if and only if y will work. For all y, if y works, then y is playing tennis. For all y, if y plays tennis, 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 if y listens to a song, then y is drinking water.
 From the fact that for all y, y will listen to a song if and only if y will work we can infer that for all y, if y listens to a song, then y is working via universal biconditional elimination. From the fact that for all y, if y works, then y is playing tennis, and that for all y, if y plays tennis, then y is climbing a mountain we can infer that for all y, if y works, then y is climbing a mountain via universal transitivity. From the fact that for all y, if y works, then y is climbing a mountain, and that there is at least one y for which if y climbs a mountain, then y is drinking water we can infer that there is at least one y for which if y works, then y is drinking water via existential transitivity. Finally, from the fact that for all y, if y listens to a song, then y is working, and that there is at least one y for which if y works, then y is drinking water we can infer that there is at least one y for which if y listens to a song, then y is drinking water via existential transitivity. Therefore, the answer is yes.