Consider the following premises. For all x_2, x_2 is a bartender if and only if x_2 will climb a mountain. For all x_2, if x_2 climbs a mountain, then x_2 is a writer. For all x_2, x_2 is a writer if and only if x_2 will climb a mountain. For all x_2, x_2 is not a writer. Can we infer the following from them? For all x_2, x_2 is not a bartender.
 From the fact that for all x_2, x_2 is a writer if and only if x_2 will climb a mountain we can infer that for all x_2, if x_2 were a writer, then x_2 is climbing a mountain. From the fact that for all x_2, if x_2 climbs a mountain, then x_2 is a writer, and that for all x_2, if x_2 were a writer, then x_2 is climbing a mountain we can infer that for all x_2, x_2 will climb a mountain if and only if x_2 is a writer. From the fact that for all x_2, x_2 will climb a mountain if and only if x_2 is a writer, and that for all x_2, x_2 is not a writer we can infer that for all x_2, x_2 will not climb a mountain. Finally, from the fact that for all x_2, x_2 is a bartender if and only if x_2 will climb a mountain, and that for all x_2, x_2 will not climb a mountain we can infer that for all x_2, x_2 is not a bartender. Therefore, the answer is yes.