Consider the following premises. For all x_9, x_9 is a carpenter if and only if x_9 will take a plane. For all x_9, x_9 will not take a plane. There is at least one x_9 for which x_9 will sleep if and only if x_9 is a carpenter. Can we infer the following from them? If we can, name the inference rule being used: There is at least one x_9 for which x_9 will not sleep.
 From the fact that for all x_9, x_9 is a carpenter if and only if x_9 will take a plane, and that for all x_9, x_9 will not take a plane we can infer that for all x_9, x_9 is not a carpenter via universal biconditional elimination. From the fact that there is at least one x_9 for which x_9 will sleep if and only if x_9 is a carpenter we can infer that there is at least one x_9 for which if x_9 sleeps, then x_9 is a carpenter via existential biconditional elimination. Finally, from the fact that there is at least one x_9 for which if x_9 sleeps, then x_9 is a carpenter, and that for all x_9, x_9 is not a carpenter we can infer that there is at least one x_9 for which x_9 will not sleep via existential modus tollens. Therefore, the answer is yes.