Consider the following premises. For all x_12, if x_12 sleeps, then x_12 is playing tennis. There is at least one x_12 for which x_12 is an office clerk or x_12 will sleep. For all x_12, if x_12 climbs a mountain, then x_12 is not playing tennis. For all x_12, x_12 is an office clerk if and only if x_12 will play tennis. 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 not climb a mountain.
 From the fact that for all x_12, x_12 is an office clerk if and only if x_12 will play tennis we can infer that for all x_12, if x_12 were an office clerk, then x_12 is playing tennis via universal biconditional elimination. From the fact that for all x_12, if x_12 were an office clerk, then x_12 is playing tennis, that for all x_12, if x_12 sleeps, then x_12 is playing tennis, and that there is at least one x_12 for which x_12 is an office clerk or x_12 will sleep we can infer that there is at least one x_12 for which x_12 will play tennis via existential disjunction elimination. Finally, from the fact that for all x_12, if x_12 climbs a mountain, then x_12 is not playing tennis, and that there is at least one x_12 for which x_12 will play tennis we can infer that there is at least one x_12 for which x_12 will not climb a mountain via existential modus tollens. Therefore, the answer is yes.