Consider the following premises. For all x_6, if x_6 writes a letter, then x_6 is sleeping. For all x_6, if x_6 sleeps, then x_6 is working. For all x_6, if x_6 works, then x_6 is taking a plane. There is at least one x_6 for which if x_6 takes a plane, then x_6 is writing a letter. Can we infer the following from them? If we can, name the inference rule being used: There is at least one x_6 for which x_6 will take a plane if and only if x_6 will sleep.
 From the fact that for all x_6, if x_6 sleeps, then x_6 is working, and that for all x_6, if x_6 works, then x_6 is taking a plane we can infer that for all x_6, if x_6 sleeps, then x_6 is taking a plane via universal transitivity. From the fact that there is at least one x_6 for which if x_6 takes a plane, then x_6 is writing a letter, and that for all x_6, if x_6 writes a letter, then x_6 is sleeping we can infer that there is at least one x_6 for which if x_6 takes a plane, then x_6 is sleeping via existential transitivity. Finally, from the fact that there is at least one x_6 for which if x_6 takes a plane, then x_6 is sleeping, and that for all x_6, if x_6 sleeps, then x_6 is taking a plane we can infer that there is at least one x_6 for which x_6 will take a plane if and only if x_6 will sleep via existential biconditional introduction. Therefore, the answer is yes.