Consider the following premises. For all x_4, x_4 is a poet if and only if x_4 will sleep. For all x_4, x_4 is not an astronaut. For all x_4, x_4 is a poet if and only if x_4 is an astronaut. Can we infer the following from them? If we can, name the inference rule being used: For all x_4, x_4 will not sleep.
 From the fact that for all x_4, x_4 is a poet if and only if x_4 is an astronaut, and that for all x_4, x_4 is not an astronaut we can infer that for all x_4, x_4 is not a poet via universal biconditional elimination. Finally, from the fact that for all x_4, x_4 is a poet if and only if x_4 will sleep, and that for all x_4, x_4 is not a poet we can infer that for all x_4, x_4 will not sleep via universal biconditional elimination. Therefore, the answer is yes.