Consider the following premises. There is at least one x_4 for which if x_4 were a bartender, then x_4 is climbing a mountain. For all x_4, if x_4 were a bartender, then x_4 is an office clerk. For all x_4, if x_4 climbs a mountain, then x_4 is a bartender. For all x_4, x_4 is an office clerk if and only if x_4 will climb a mountain. Can we infer the following from them? There is at least one x_4 for which x_4 will climb a mountain if and only if x_4 is a bartender.
 From the fact that there is at least one x_4 for which if x_4 were a bartender, then x_4 is climbing a mountain, and that for all x_4, if x_4 climbs a mountain, then x_4 is a bartender we can infer that there is at least one x_4 for which x_4 is a bartender if and only if x_4 will climb a mountain. From the fact that there is at least one x_4 for which x_4 is a bartender if and only if x_4 will climb a mountain we can infer that there is at least one x_4 for which if x_4 climbs a mountain, then x_4 is a bartender. From the fact that for all x_4, x_4 is an office clerk if and only if x_4 will climb a mountain we can infer that for all x_4, if x_4 were an office clerk, then x_4 is climbing a mountain. From the fact that for all x_4, if x_4 were a bartender, then x_4 is an office clerk, and that for all x_4, if x_4 were an office clerk, then x_4 is climbing a mountain we can infer that for all x_4, if x_4 were a bartender, then x_4 is climbing a mountain. Finally, from the fact that there is at least one x_4 for which if x_4 climbs a mountain, then x_4 is a bartender, and that for all x_4, if x_4 were a bartender, then x_4 is climbing a mountain we can infer that there is at least one x_4 for which x_4 will climb a mountain if and only if x_4 is a bartender. Therefore, the answer is yes.