Consider the following premises. There is at least one z for which z will not cook. For all z, z will drink water or z will not cook. For all z, z will cook or z will not drink water. Can we infer the following from them? There is at least one z for which z will not drink water.
 From the fact that for all z, z will drink water or z will not cook we can infer that for all z, if z cooks, then z is drinking water. From the fact that for all z, z will cook or z will not drink water we can infer that for all z, if z drinks water, then z is cooking. From the fact that for all z, if z cooks, then z is drinking water, and that for all z, if z drinks water, then z is cooking we can infer that for all z, z will cook if and only if z will drink water. Finally, from the fact that for all z, z will cook if and only if z will drink water, and that there is at least one z for which z will not cook we can infer that there is at least one z for which z will not drink water. Therefore, the answer is yes.