Consider the following premises. For all y, if y sleeps, then y is a carpenter. There is at least one y for which if y sleeps, then y is a carpenter. For all y, y is a carpenter if and only if y will sleep. Can we infer the following from them? There is at least one y for which y will sleep if and only if y is a carpenter.
 From the fact that for all y, y is a carpenter if and only if y will sleep we can infer that for all y, if y were a carpenter, then y is sleeping. From the fact that for all y, if y were a carpenter, then y is sleeping, and that for all y, if y sleeps, then y is a carpenter we can infer that for all y, y is a carpenter if and only if y will sleep. From the fact that for all y, y is a carpenter if and only if y will sleep we can infer that for all y, if y were a carpenter, then y is sleeping. Finally, from the fact that there is at least one y for which if y sleeps, then y is a carpenter, and that for all y, if y were a carpenter, then y is sleeping we can infer that there is at least one y for which y will sleep if and only if y is a carpenter. Therefore, the answer is yes.