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