Consider the following premises. There is at least one z_4 for which if z_4 were a carpenter, then z_4 is working. For all z_4, if z_4 works, then z_4 is a carpenter. For all z_4, z_4 is a mechanic if and only if z_4 will work. For all z_4, if z_4 were a carpenter, then z_4 is a mechanic. Can we infer the following from them? There is at least one z_4 for which z_4 will work if and only if z_4 is a carpenter.
 From the fact that there is at least one z_4 for which if z_4 were a carpenter, then z_4 is working, and that for all z_4, if z_4 works, then z_4 is a carpenter we can infer that there is at least one z_4 for which z_4 is a carpenter if and only if z_4 will work. From the fact that there is at least one z_4 for which z_4 is a carpenter if and only if z_4 will work we can infer that there is at least one z_4 for which if z_4 works, then z_4 is a carpenter. From the fact that for all z_4, z_4 is a mechanic if and only if z_4 will work we can infer that for all z_4, if z_4 were a mechanic, then z_4 is working. From the fact that for all z_4, if z_4 were a carpenter, then z_4 is a mechanic, and that for all z_4, if z_4 were a mechanic, then z_4 is working we can infer that for all z_4, if z_4 were a carpenter, then z_4 is working. Finally, from the fact that there is at least one z_4 for which if z_4 works, then z_4 is a carpenter, and that for all z_4, if z_4 were a carpenter, then z_4 is working we can infer that there is at least one z_4 for which z_4 will work if and only if z_4 is a carpenter. Therefore, the answer is yes.