Consider the following premises. For all x_9, if x_9 climbs a mountain, then x_9 is a bartender. For all x_9, x_9 is not a bartender. For all x_9, x_9 is excited if and only if x_9 will climb a mountain. There is at least one x_9 for which if x_9 were a bartender, then x_9 is climbing a mountain. Can we infer the following from them? If we can, name the inference rule being used: For all x_9, x_9 is excited.
 From the fact that there is at least one x_9 for which if x_9 were a bartender, then x_9 is climbing a mountain, and that for all x_9, if x_9 climbs a mountain, then x_9 is a bartender we can infer that there is at least one x_9 for which x_9 is a bartender if and only if x_9 will climb a mountain via existential biconditional introduction. From the fact that there is at least one x_9 for which x_9 is a bartender if and only if x_9 will climb a mountain, and that for all x_9, x_9 is not a bartender we can infer that there is at least one x_9 for which x_9 will not climb a mountain via existential biconditional elimination. Finally, from the fact that for all x_9, x_9 is excited if and only if x_9 will climb a mountain, and that there is at least one x_9 for which x_9 will not climb a mountain we can infer that there is at least one x_9 for which x_9 is not excited via existential biconditional elimination, which contradicts that For all x_9, x_9 is excited. Therefore, the answer is no.