Consider the following premises. For all y, y will climb a mountain. For all y, y will climb a mountain if and only if y is fast. There is at least one y for which if y reads a book, then y is fast. 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 fast.
 From the fact that for all y, y will climb a mountain if and only if y is fast we can infer that for all y, if y climbs a mountain, then y is fast via universal biconditional elimination. From the fact that for all y, y will climb a mountain we can infer that for all y, y will climb a mountain or y will read a book via universal addition. Finally, from the fact that for all y, if y climbs a mountain, then y is fast, that there is at least one y for which if y reads a book, then y is fast, and that for all y, y will climb a mountain or y will read a book we can infer that there is at least one y for which y is fast via existential disjunction elimination. Therefore, the answer is yes.