Consider the following premises. For all x_10, if x_10 sleeps, then x_10 is climbing a mountain. For all x_10, if x_10 climbs a mountain, then x_10 is sleeping. If Elisabeth climbs a mountain, then it is late. Can we infer the following from them? If we can, name the inference rule being used: If Elisabeth sleeps, then it is late.
 From the fact that for all x_10, if x_10 climbs a mountain, then x_10 is sleeping, and that for all x_10, if x_10 sleeps, then x_10 is climbing a mountain we can infer that for all x_10, x_10 will climb a mountain if and only if x_10 will sleep via universal biconditional introduction. From the fact that for all x_10, x_10 will climb a mountain if and only if x_10 will sleep we can infer that for all x_10, if x_10 sleeps, then x_10 is climbing a mountain via universal biconditional elimination. From the fact that for all x_10, if x_10 sleeps, then x_10 is climbing a mountain we can infer that if Elisabeth sleeps, then Elisabeth is climbing a mountain via universal instantiation. Finally, from the fact that if Elisabeth sleeps, then Elisabeth is climbing a mountain, and that if Elisabeth climbs a mountain, then it is late we can infer that if Elisabeth sleeps, then it is late via transitivity. Therefore, the answer is yes.