Consider the following premises. There is at least one z_3 for which z_3 is a soldier if and only if z_3 will play tennis. For all z_3, z_3 will play tennis or z_3 will not climb a mountain. For all z_3, z_3 will climb a mountain or z_3 is a soldier. Can we infer the following from them? If we can, name the inference rule being used: For all z_3, z_3 will not play tennis.
 From the fact that for all z_3, z_3 will play tennis or z_3 will not climb a mountain we can infer that for all z_3, if z_3 climbs a mountain, then z_3 is playing tennis via universal rewriting. From the fact that there is at least one z_3 for which z_3 is a soldier if and only if z_3 will play tennis we can infer that there is at least one z_3 for which if z_3 were a soldier, then z_3 is playing tennis via existential biconditional elimination. Finally, from the fact that for all z_3, if z_3 climbs a mountain, then z_3 is playing tennis, that there is at least one z_3 for which if z_3 were a soldier, then z_3 is playing tennis, and that for all z_3, z_3 will climb a mountain or z_3 is a soldier we can infer that there is at least one z_3 for which z_3 will play tennis via existential disjunction elimination, which contradicts that For all z_3, z_3 will not play tennis. Therefore, the answer is no.