Consider the following premises. For all x_3, x_3 will not sleep. For all x_3, x_3 will cook if and only if x_3 will sleep. Can we infer the following from them? If we can, name the inference rule being used: There is at least one x_3 for which x_3 will cook.
 From the fact that for all x_3, x_3 will cook if and only if x_3 will sleep we can infer that for all x_3, if x_3 cooks, then x_3 is sleeping via universal biconditional elimination. Finally, from the fact that for all x_3, if x_3 cooks, then x_3 is sleeping, and that for all x_3, x_3 will not sleep we can infer that for all x_3, x_3 will not cook via universal modus tollens, which contradicts that There is at least one x_3 for which x_3 will cook. Therefore, the answer is no.