Consider the following premises. For all x_9, if x_9 were a doctor, then x_9 is drinking water. For all x_9, if x_9 were a doctor, then x_9 is not climbing a mountain. There is at least one x_9 for which x_9 will climb a mountain. For all x_9, if x_9 drinks water, then x_9 is a doctor. 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 drink water.
 From the fact that for all x_9, if x_9 were a doctor, then x_9 is not climbing a mountain, and that there is at least one x_9 for which x_9 will climb a mountain we can infer that there is at least one x_9 for which x_9 is not a doctor via existential modus tollens. From the fact that for all x_9, if x_9 were a doctor, then x_9 is drinking water, and that for all x_9, if x_9 drinks water, then x_9 is a doctor we can infer that for all x_9, x_9 is a doctor if and only if x_9 will drink water via universal biconditional introduction. From the fact that for all x_9, x_9 is a doctor if and only if x_9 will drink water we can infer that for all x_9, if x_9 drinks water, then x_9 is a doctor via universal biconditional elimination. Finally, from the fact that for all x_9, if x_9 drinks water, then 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 drink water via existential modus tollens. Therefore, the answer is yes.