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