Consider the following premises. For all x, if x were an astronaut, then x is listening to a song. For all x, if x listens to a song, then x is a taxi driver. There is at least one x for which x is an astronaut if and only if x is a taxi driver. Can we infer the following from them? If we can, name the inference rule being used: There is at least one x for which x is a taxi driver if and only if x is an astronaut.
 From the fact that there is at least one x for which x is an astronaut if and only if x is a taxi driver we can infer that there is at least one x for which if x were a taxi driver, then x is an astronaut via existential biconditional elimination. From the fact that for all x, if x were an astronaut, then x is listening to a song, and that for all x, if x listens to a song, then x is a taxi driver we can infer that for all x, if x were an astronaut, then x is a taxi driver via universal transitivity. Finally, from the fact that there is at least one x for which if x were a taxi driver, then x is an astronaut, and that for all x, if x were an astronaut, then x is a taxi driver we can infer that there is at least one x for which x is a taxi driver if and only if x is an astronaut via existential biconditional introduction. Therefore, the answer is yes.