Consider the following premises. For all x_6, if x_6 climbs a mountain, then x_6 is a bartender. For all x_6, if x_6 were a nurse, then x_6 is climbing a mountain. For all x_6, x_6 is a nurse or x_6 is not a bartender. Can we infer the following from them? If we can, name the inference rule being used: There is at least one x_6 for which if is not true that x_6 is a bartender if and only if x_6 will climb a mountain.
 From the fact that for all x_6, x_6 is a nurse or x_6 is not a bartender we can infer that for all x_6, if x_6 were a bartender, then x_6 is a nurse via universal rewriting. From the fact that for all x_6, if x_6 were a bartender, then x_6 is a nurse, and that for all x_6, if x_6 were a nurse, then x_6 is climbing a mountain we can infer that for all x_6, if x_6 were a bartender, then x_6 is climbing a mountain via universal transitivity. Finally, from the fact that for all x_6, if x_6 were a bartender, then x_6 is climbing a mountain, and that for all x_6, if x_6 climbs a mountain, then x_6 is a bartender we can infer that for all x_6, x_6 is a bartender if and only if x_6 will climb a mountain via universal biconditional introduction, which contradicts that There is at least one x_6 for which if is not true that x_6 is a bartender if and only if x_6 will climb a mountain. Therefore, the answer is no.