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