Consider the following premises. For all x_12, if x_12 were a nurse, then x_12 is excited. For all x_12, x_12 is poor if and only if x_12 is excited. For all x_12, x_12 is not poor. For all x_12, if x_12 cooks, then x_12 is a nurse. Can we infer the following from them? If we can, name the inference rule being used: There is at least one x_12 for which x_12 will cook.
 From the fact that for all x_12, x_12 is poor if and only if x_12 is excited we can infer that for all x_12, if x_12 were excited, then x_12 is poor via universal biconditional elimination. From the fact that for all x_12, if x_12 were a nurse, then x_12 is excited, and that for all x_12, if x_12 were excited, then x_12 is poor we can infer that for all x_12, if x_12 were a nurse, then x_12 is poor via universal transitivity. From the fact that for all x_12, if x_12 cooks, then x_12 is a nurse, and that for all x_12, if x_12 were a nurse, then x_12 is poor we can infer that for all x_12, if x_12 cooks, then x_12 is poor via universal transitivity. Finally, from the fact that for all x_12, if x_12 cooks, then x_12 is poor, and that for all x_12, x_12 is not poor we can infer that for all x_12, x_12 will not cook via universal modus tollens, which contradicts that There is at least one x_12 for which x_12 will cook. Therefore, the answer is no.