Consider the following premises. For all x_9, x_9 will not sleep. There is at least one x_9 for which x_9 is a doctor if and only if x_9 will climb a mountain. For all x_9, x_9 will play a game if and only if x_9 is a doctor. For all x_9, if x_9 climbs a mountain, then x_9 is sleeping. Can we infer the following from them? If we can, name the inference rule being used: There is at least one x_9 for which x_9 will not play a game.
 From the fact that for all x_9, if x_9 climbs a mountain, then x_9 is sleeping, and that for all x_9, x_9 will not sleep we can infer that for all x_9, x_9 will not climb a mountain via universal modus tollens. From the fact that there is at least one x_9 for which x_9 is a doctor if and only if x_9 will climb a mountain, and that for all x_9, x_9 will not climb a mountain we can infer that there is at least one x_9 for which x_9 is not a doctor via existential biconditional elimination. Finally, from the fact that for all x_9, x_9 will play a game if and only if x_9 is a doctor, and that there is at least one x_9 for which x_9 is not a doctor we can infer that there is at least one x_9 for which x_9 will not play a game via existential biconditional elimination. Therefore, the answer is yes.