Consider the following premises. For all y, if y climbs a mountain, then y is sleeping. For all y, if y climbs a mountain, then y is a janitor. There is at least one y for which y is a janitor or y will not sleep. For all y, if y were a janitor, then y is climbing a mountain. Can we infer the following from them? If we can, name the inference rule being used: For all y, if is not true that y is a janitor if and only if y will sleep.
 From the fact that there is at least one y for which y is a janitor or y will not sleep we can infer that there is at least one y for which if y sleeps, then y is a janitor via existential rewriting. Finally, from the fact that for all y, if y were a janitor, then y is climbing a mountain, and that for all y, if y climbs a mountain, then y is a janitor we can infer that for all y, y is a janitor if and only if y will climb a mountain via universal biconditional introduction, which contradicts that For all y, if is not true that y is a janitor if and only if y will sleep. Therefore, the answer is no.