Consider the following premises. For all y, if y were a musician, then y is working. There is at least one y for which y is excited. For all y, y will cook or y is a musician. For all y, if y cooks, then y is working. 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 is excited and y will work.
 From the fact that for all y, if y cooks, then y is working, that for all y, if y were a musician, then y is working, and that for all y, y will cook or y is a musician we can infer that for all y, y will work via universal disjunction elimination. Finally, from the fact that there is at least one y for which y is excited, and that for all y, y will work we can infer that there is at least one y for which y is excited and y will work via existential conjunction. Therefore, the answer is yes.