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