Consider the following premises. For all y_4, if y_4 sleeps, then y_4 is climbing a mountain. For all y_4, y_4 will not climb a mountain. There is at least one y_4 for which if y_4 climbs a mountain, then y_4 is sleeping. Can we infer the following from them? If we can, name the inference rule being used: For all y_4, y_4 will sleep.
 From the fact that for all y_4, if y_4 sleeps, then y_4 is climbing a mountain, and that there is at least one y_4 for which if y_4 climbs a mountain, then y_4 is sleeping we can infer that there is at least one y_4 for which y_4 will sleep if and only if y_4 will climb a mountain via existential biconditional introduction. From the fact that there is at least one y_4 for which y_4 will sleep if and only if y_4 will climb a mountain we can infer that there is at least one y_4 for which if y_4 sleeps, then y_4 is climbing a mountain via existential biconditional elimination. Finally, from the fact that there is at least one y_4 for which if y_4 sleeps, then y_4 is climbing a mountain, and that for all y_4, y_4 will not climb a mountain we can infer that there is at least one y_4 for which y_4 will not sleep via existential modus tollens, which contradicts that For all y_4, y_4 will sleep. Therefore, the answer is no.