Consider the following premises. For all z, z is happy or z will not climb a mountain. For all z, z will climb a mountain or z is not happy. For all z, z will not climb a mountain. 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 is happy.
 From the fact that for all z, z is happy or z will not climb a mountain we can infer that for all z, if z climbs a mountain, then z is happy via universal rewriting. From the fact that for all z, z will climb a mountain or z is not happy we can infer that for all z, if z were happy, then z is climbing a mountain via universal rewriting. From the fact that for all z, if z climbs a mountain, then z is happy, and that for all z, if z were happy, then z is climbing a mountain we can infer that for all z, z will climb a mountain if and only if z is happy via universal biconditional introduction. Finally, from the fact that for all z, z will climb a mountain if and only if z is happy, and that for all z, z will not climb a mountain we can infer that for all z, z is not happy via universal biconditional elimination, which contradicts that There is at least one z for which z is happy. Therefore, the answer is no.