Consider the following premises. For all y, if y were a scientist, then y is a musician. For all y, y is an astronaut or y is not a musician. There is at least one y for which if y were an astronaut, then y is a scientist. Can we infer the following from them? There is at least one y for which y is a musician if and only if y is an astronaut.
 From the fact that for all y, y is an astronaut or y is not a musician we can infer that for all y, if y were a musician, then y is an astronaut. From the fact that there is at least one y for which if y were an astronaut, then y is a scientist, and that for all y, if y were a scientist, then y is a musician we can infer that there is at least one y for which if y were an astronaut, then y is a musician. Finally, from the fact that for all y, if y were a musician, then y is an astronaut, and that there is at least one y for which if y were an astronaut, then y is a musician we can infer that there is at least one y for which y is a musician if and only if y is an astronaut. Therefore, the answer is yes.