Consider the following premises. There is at least one z_4 for which z_4 will work or z_4 will not climb a mountain. For all z_4, z_4 will climb a mountain or z_4 will not work. Can we infer the following from them? There is at least one z_4 for which z_4 will work if and only if z_4 will climb a mountain.
 From the fact that for all z_4, z_4 will climb a mountain or z_4 will not work we can infer that for all z_4, if z_4 works, then z_4 is climbing a mountain. From the fact that there is at least one z_4 for which z_4 will work or z_4 will not climb a mountain we can infer that there is at least one z_4 for which if z_4 climbs a mountain, then z_4 is working. Finally, from the fact that for all z_4, if z_4 works, then z_4 is climbing a mountain, and that there is at least one z_4 for which if z_4 climbs a mountain, then z_4 is working we can infer that there is at least one z_4 for which z_4 will work if and only if z_4 will climb a mountain. Therefore, the answer is yes.