Consider the following premises. For all z, if z cooks, then z is happy. For all z, if z were happy, then z is cooking. For all z, z is a nurse if and only if z will cook. For all z, z is not a nurse. Can we infer the following from them? If we can, name the inference rule being used: There is at least one z for which z is happy.
 From the fact that for all z, z is a nurse if and only if z will cook, and that for all z, z is not a nurse we can infer that for all z, z will not cook via universal biconditional elimination. From the fact that for all z, if z were happy, then z is cooking, and that for all z, if z cooks, then z is happy we can infer that for all z, z is happy if and only if z will cook via universal biconditional introduction. Finally, from the fact that for all z, z is happy if and only if z will cook, and that for all z, z will not cook we can infer that for all z, z is not happy via universal biconditional elimination, which contradicts that There is at least one z for which z is happy. Therefore, the answer is no.