Consider the following premises. For all z_1, if z_1 were bored, then z_1 is climbing a mountain. For all z_1, z_1 will climb a mountain if and only if z_1 is bored. For all z_1, if z_1 climbs a mountain, then z_1 is bored. Can we infer the following from them? If we can, name the inference rule being used: There is at least one z_1 for which if is not true that z_1 is bored if and only if z_1 will climb a mountain.
 From the fact that for all z_1, if z_1 climbs a mountain, then z_1 is bored, and that for all z_1, if z_1 were bored, then z_1 is climbing a mountain we can infer that for all z_1, z_1 will climb a mountain if and only if z_1 is bored via universal biconditional introduction. From the fact that for all z_1, z_1 will climb a mountain if and only if z_1 is bored we can infer that for all z_1, if z_1 climbs a mountain, then z_1 is bored via universal biconditional elimination. From the fact that for all z_1, z_1 will climb a mountain if and only if z_1 is bored we can infer that for all z_1, if z_1 were bored, then z_1 is climbing a mountain via universal biconditional elimination. Finally, from the fact that for all z_1, if z_1 were bored, then z_1 is climbing a mountain, and that for all z_1, if z_1 climbs a mountain, then z_1 is bored we can infer that for all z_1, z_1 is bored if and only if z_1 will climb a mountain via universal biconditional introduction, which contradicts that There is at least one z_1 for which if is not true that z_1 is bored if and only if z_1 will climb a mountain. Therefore, the answer is no.