Consider the following premises. There is at least one z_2 for which if z_2 sleeps, then z_2 is cooking. For all z_2, if z_2 cooks, then z_2 is sleeping. For all z_2, if z_2 cooks, then z_2 is sleeping. Can we infer the following from them? For all z_2, if is not true that z_2 will sleep if and only if z_2 will cook.
 From the fact that for all z_2, if z_2 cooks, then z_2 is sleeping, and that there is at least one z_2 for which if z_2 sleeps, then z_2 is cooking we can infer that there is at least one z_2 for which z_2 will cook if and only if z_2 will sleep. From the fact that there is at least one z_2 for which z_2 will cook if and only if z_2 will sleep we can infer that there is at least one z_2 for which if z_2 sleeps, then z_2 is cooking. Finally, from the fact that there is at least one z_2 for which if z_2 sleeps, then z_2 is cooking, and that for all z_2, if z_2 cooks, then z_2 is sleeping we can infer that there is at least one z_2 for which z_2 will sleep if and only if z_2 will cook, which contradicts that For all z_2, if is not true that z_2 will sleep if and only if z_2 will cook. Therefore, the answer is no.