Consider the following premises. For all x, x is a janitor if and only if x will sleep. For all x, if x were a janitor, then x is sleeping. There is at least one x for which x will not sleep. Can we infer the following from them? If we can, name the inference rule being used: There is at least one x for which x is not a janitor.
 From the fact that for all x, x is a janitor if and only if x will sleep we can infer that for all x, if x sleeps, then x is a janitor via universal biconditional elimination. From the fact that for all x, if x were a janitor, then x is sleeping, and that for all x, if x sleeps, then x is a janitor we can infer that for all x, x is a janitor if and only if x will sleep via universal biconditional introduction. Finally, from the fact that for all x, x is a janitor if and only if x will sleep, and that there is at least one x for which x will not sleep we can infer that there is at least one x for which x is not a janitor via existential biconditional elimination. Therefore, the answer is yes.