Consider the following premises. There is at least one z for which z will climb a mountain or z will not go running. For all z, z is an astronaut if and only if z will climb a mountain. For all z, z is an astronaut or z will go running. Can we infer the following from them? If we can, name the inference rule being used: For all z, z will not climb a mountain.
 From the fact that there is at least one z for which z will climb a mountain or z will not go running we can infer that there is at least one z for which if z goes running, then z is climbing a mountain via existential rewriting. From the fact that for all z, z is an astronaut if and only if z will climb a mountain we can infer that for all z, if z were an astronaut, then z is climbing a mountain via universal biconditional elimination. Finally, from the fact that for all z, if z were an astronaut, then z is climbing a mountain, that there is at least one z for which if z goes running, then z is climbing a mountain, and that for all z, z is an astronaut or z will go running we can infer that there is at least one z for which z will climb a mountain via existential disjunction elimination, which contradicts that For all z, z will not climb a mountain. Therefore, the answer is no.