Consider the following premises. For all x_4, x_4 will cook if and only if x_4 is a janitor. For all x_4, if x_4 cooks, then x_4 is a janitor. Can we infer the following from them? If we can, name the inference rule being used: There is at least one x_4 for which if is not true that x_4 is a janitor if and only if x_4 will cook.
 From the fact that for all x_4, x_4 will cook if and only if x_4 is a janitor we can infer that for all x_4, if x_4 were a janitor, then x_4 is cooking via universal biconditional elimination. Finally, from the fact that for all x_4, if x_4 were a janitor, then x_4 is cooking, and that for all x_4, if x_4 cooks, then x_4 is a janitor we can infer that for all x_4, x_4 is a janitor if and only if x_4 will cook via universal biconditional introduction, which contradicts that There is at least one x_4 for which if is not true that x_4 is a janitor if and only if x_4 will cook. Therefore, the answer is no.