Consider the following premises. There is at least one x for which x is an astronaut if and only if x is an artist. For all x, x is an astronaut if and only if x is an artist. Can we infer the following from them? There is at least one x for which x is an artist if and only if x is an astronaut.
 From the fact that for all x, x is an astronaut if and only if x is an artist we can infer that for all x, if x were an astronaut, then x is an artist. From the fact that there is at least one x for which x is an astronaut if and only if x is an artist we can infer that there is at least one x for which if x were an artist, then x is an astronaut. Finally, from the fact that there is at least one x for which if x were an artist, then x is an astronaut, and that for all x, if x were an astronaut, then x is an artist we can infer that there is at least one x for which x is an artist if and only if x is an astronaut. Therefore, the answer is yes.