Consider the following premises. For all z, if z makes tea, then z is an astronaut. There is at least one z for which z is an astronaut or z is not fast. For all z, z will cook or z will make tea. For all z, if z cooks, then z is fast. Can we infer the following from them? If we can, name the inference rule being used: For all z, z is not an astronaut.
 From the fact that there is at least one z for which z is an astronaut or z is not fast we can infer that there is at least one z for which if z were fast, then z is an astronaut via existential rewriting. From the fact that for all z, if z cooks, then z is fast, and that there is at least one z for which if z were fast, then z is an astronaut we can infer that there is at least one z for which if z cooks, then z is an astronaut via existential transitivity. Finally, from the fact that there is at least one z for which if z cooks, then z is an astronaut, that for all z, if z makes tea, then z is an astronaut, and that for all z, z will cook or z will make tea we can infer that there is at least one z for which z is an astronaut via existential disjunction elimination, which contradicts that For all z, z is not an astronaut. Therefore, the answer is no.