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