Consider the following premises. For all y_2, y_2 will climb a mountain if and only if y_2 will sleep. For all y_2, y_2 will climb a mountain or y_2 will not sleep. Can we infer the following from them? If we can, name the inference rule being used: There is at least one y_2 for which if is not true that y_2 will sleep if and only if y_2 will climb a mountain.
 From the fact that for all y_2, y_2 will climb a mountain if and only if y_2 will sleep we can infer that for all y_2, if y_2 climbs a mountain, then y_2 is sleeping via universal biconditional elimination. From the fact that for all y_2, y_2 will climb a mountain or y_2 will not sleep we can infer that for all y_2, if y_2 sleeps, then y_2 is climbing a mountain via universal rewriting. Finally, from the fact that for all y_2, if y_2 sleeps, then y_2 is climbing a mountain, and that for all y_2, if y_2 climbs a mountain, then y_2 is sleeping we can infer that for all y_2, y_2 will sleep if and only if y_2 will climb a mountain via universal biconditional introduction, which contradicts that There is at least one y_2 for which if is not true that y_2 will sleep if and only if y_2 will climb a mountain. Therefore, the answer is no.