Consider the following premises. For all x_8, x_8 will sleep if and only if x_8 will climb a mountain. There is at least one x_8 for which if x_8 climbs a mountain, then x_8 is a musician. For all x_8, if x_8 were a musician, then x_8 is a carpenter. Can we infer the following from them? If we can, name the inference rule being used: There is at least one x_8 for which if x_8 sleeps, then x_8 is a carpenter.
 From the fact that for all x_8, x_8 will sleep if and only if x_8 will climb a mountain we can infer that for all x_8, if x_8 sleeps, then x_8 is climbing a mountain via universal biconditional elimination. From the fact that for all x_8, if x_8 sleeps, then x_8 is climbing a mountain, and that there is at least one x_8 for which if x_8 climbs a mountain, then x_8 is a musician we can infer that there is at least one x_8 for which if x_8 sleeps, then x_8 is a musician via existential transitivity. Finally, from the fact that there is at least one x_8 for which if x_8 sleeps, then x_8 is a musician, and that for all x_8, if x_8 were a musician, then x_8 is a carpenter we can infer that there is at least one x_8 for which if x_8 sleeps, then x_8 is a carpenter via existential transitivity. Therefore, the answer is yes.