Consider the following premises. There is at least one x_6 for which if x_6 were a sailor, then x_6 is an astronaut. For all x_6, if x_6 cooks, then x_6 is a sailor. For all x_6, x_6 will play tennis or x_6 will cook. For all x_6, if x_6 were fast, then x_6 is an astronaut. For all x_6, if x_6 plays tennis, then x_6 is a sailor. Can we infer the following from them? If we can, name the inference rule being used: For all x_6, x_6 is not an astronaut.
 From the fact that for all x_6, if x_6 plays tennis, then x_6 is a sailor, that for all x_6, if x_6 cooks, then x_6 is a sailor, and that for all x_6, x_6 will play tennis or x_6 will cook we can infer that for all x_6, x_6 is a sailor via universal disjunction elimination. From the fact that for all x_6, x_6 is a sailor we can infer that for all x_6, x_6 is a sailor or x_6 is fast via universal addition. Finally, from the fact that there is at least one x_6 for which if x_6 were a sailor, then x_6 is an astronaut, that for all x_6, if x_6 were fast, then x_6 is an astronaut, and that for all x_6, x_6 is a sailor or x_6 is fast we can infer that there is at least one x_6 for which x_6 is an astronaut via existential disjunction elimination, which contradicts that For all x_6, x_6 is not an astronaut. Therefore, the answer is no.