Consider the following premises. For all x_5, if x_5 were a doctor, then x_5 is not climbing a mountain. For all x_5, x_5 is a cashier or x_5 will climb a mountain. For all x_5, if x_5 were a cashier, then x_5 is a scientist. For all x_5, x_5 is not a scientist. Can we infer the following from them? If we can, name the inference rule being used: For all x_5, x_5 is not a doctor.
 From the fact that for all x_5, if x_5 were a cashier, then x_5 is a scientist, and that for all x_5, x_5 is not a scientist we can infer that for all x_5, x_5 is not a cashier via universal modus tollens. From the fact that for all x_5, x_5 is a cashier or x_5 will climb a mountain, and that for all x_5, x_5 is not a cashier we can infer that for all x_5, x_5 will climb a mountain via universal disjunctive syllogism. Finally, from the fact that for all x_5, if x_5 were a doctor, then x_5 is not climbing a mountain, and that for all x_5, x_5 will climb a mountain we can infer that for all x_5, x_5 is not a doctor via universal modus tollens. Therefore, the answer is yes.