Consider the following premises. For all x_2, if x_2 takes a plane, then x_2 is an astronaut. For all x_2, if x_2 sleeps, then x_2 is taking a plane. For all x_2, if x_2 were an astronaut, then x_2 is sleeping. For all x_2, x_2 is not an astronaut. Can we infer the following from them? If we can, name the inference rule being used: There is at least one x_2 for which x_2 will sleep.
 From the fact that for all x_2, if x_2 sleeps, then x_2 is taking a plane, and that for all x_2, if x_2 takes a plane, then x_2 is an astronaut we can infer that for all x_2, if x_2 sleeps, then x_2 is an astronaut via universal transitivity. From the fact that for all x_2, if x_2 were an astronaut, then x_2 is sleeping, and that for all x_2, if x_2 sleeps, then x_2 is an astronaut we can infer that for all x_2, x_2 is an astronaut if and only if x_2 will sleep via universal biconditional introduction. Finally, from the fact that for all x_2, x_2 is an astronaut if and only if x_2 will sleep, and that for all x_2, x_2 is not an astronaut we can infer that for all x_2, x_2 will not sleep via universal biconditional elimination, which contradicts that There is at least one x_2 for which x_2 will sleep. Therefore, the answer is no.