Consider the following premises. For all x_12, x_12 will cook or x_12 will climb a mountain. For all x_12, x_12 is a bartender or x_12 will sleep. There is at least one x_12 for which if x_12 climbs a mountain, then x_12 is a carpenter. For all x_12, if x_12 cooks, then x_12 is a carpenter. For all x_12, x_12 is not a bartender. Can we infer the following from them? If we can, name the inference rule being used: There is at least one x_12 for which x_12 will sleep and x_12 is a carpenter.
 From the fact that for all x_12, if x_12 cooks, then x_12 is a carpenter, that there is at least one x_12 for which if x_12 climbs a mountain, then x_12 is a carpenter, and that for all x_12, x_12 will cook or x_12 will climb a mountain we can infer that there is at least one x_12 for which x_12 is a carpenter via existential disjunction elimination. From the fact that for all x_12, x_12 is a bartender or x_12 will sleep, and that for all x_12, x_12 is not a bartender we can infer that for all x_12, x_12 will sleep via universal disjunctive syllogism. Finally, from the fact that for all x_12, x_12 will sleep, and that there is at least one x_12 for which x_12 is a carpenter we can infer that there is at least one x_12 for which x_12 will sleep and x_12 is a carpenter via existential conjunction. Therefore, the answer is yes.