Consider the following premises. There is at least one z for which if z takes a plane, then z is sleeping. For all z, z is sad if and only if z will sleep. For all z, z will take a plane or z is not sad. Can we infer the following from them? If we can, name the inference rule being used: There is at least one z for which z will take a plane if and only if z is sad.
 From the fact that for all z, z will take a plane or z is not sad we can infer that for all z, if z were sad, then z is taking a plane via universal rewriting. From the fact that for all z, z is sad if and only if z will sleep we can infer that for all z, if z sleeps, then z is sad via universal biconditional elimination. From the fact that there is at least one z for which if z takes a plane, then z is sleeping, and that for all z, if z sleeps, then z is sad we can infer that there is at least one z for which if z takes a plane, then z is sad via existential transitivity. Finally, from the fact that there is at least one z for which if z takes a plane, then z is sad, and that for all z, if z were sad, then z is taking a plane we can infer that there is at least one z for which z will take a plane if and only if z is sad via existential biconditional introduction. Therefore, the answer is yes.