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