Consider the following premises. There is at least one x_6 for which x_6 will climb a mountain. For all x_6, if x_6 were a janitor, then x_6 is cooking. For all x_6, x_6 will climb a mountain if and only if x_6 will cook. Can we infer the following from them? If we can, name the inference rule being used: For all x_6, x_6 will not cook.
 From the fact that for all x_6, x_6 will climb a mountain if and only if x_6 will cook we can infer that for all x_6, if x_6 climbs a mountain, then x_6 is cooking via universal biconditional elimination. From the fact that there is at least one x_6 for which x_6 will climb a mountain we can infer that there is at least one x_6 for which x_6 will climb a mountain or x_6 is a janitor via existential addition. Finally, from the fact that for all x_6, if x_6 climbs a mountain, then x_6 is cooking, that for all x_6, if x_6 were a janitor, then x_6 is cooking, and that there is at least one x_6 for which x_6 will climb a mountain or x_6 is a janitor we can infer that there is at least one x_6 for which x_6 will cook via existential disjunction elimination, which contradicts that For all x_6, x_6 will not cook. Therefore, the answer is no.