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