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