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