Consider the following premises. There is at least one z for which z is a janitor if and only if z will cook. For all z, z will cook or z is not a janitor. Can we infer the following from them? For all z, if is not true that z will cook if and only if z is a janitor.
 From the fact that for all z, z will cook or z is not a janitor we can infer that for all z, if z were a janitor, then z is cooking. From the fact that there is at least one z for which z is a janitor if and only if z will cook we can infer that there is at least one z for which if z cooks, then z is a janitor. Finally, from the fact that there is at least one z for which if z cooks, then z is a janitor, and that for all z, if z were a janitor, then z is cooking we can infer that there is at least one z for which z will cook if and only if z is a janitor, which contradicts that For all z, if is not true that z will cook if and only if z is a janitor. Therefore, the answer is no.