Consider the following premises. For all z, z is fast or z will not work. For all z, z is a carpenter if and only if z is fast. For all z, if z were fast, then z is working. There is at least one z for which z will not work. Can we infer the following from them? If we can, name the inference rule being used: There is at least one z for which z is not a carpenter.
 From the fact that for all z, z is fast or z will not work we can infer that for all z, if z works, then z is fast via universal rewriting. From the fact that for all z, if z were fast, then z is working, and that for all z, if z works, then z is fast we can infer that for all z, z is fast if and only if z will work via universal biconditional introduction. From the fact that for all z, z is fast if and only if z will work, and that there is at least one z for which z will not work we can infer that there is at least one z for which z is not fast via existential biconditional elimination. Finally, from the fact that for all z, z is a carpenter if and only if z is fast, and that there is at least one z for which z is not fast we can infer that there is at least one z for which z is not a carpenter via existential biconditional elimination. Therefore, the answer is yes.