Consider the following premises. For all z, if z were a politician, then z is a carpenter. For all z, if z were a carpenter, then z is a politician. For all z, z is a politician if and only if z is a carpenter. Can we infer the following from them? If we can, name the inference rule being used: There is at least one z for which the fact that z were a carpenter does not imply that z is a politician.
 From the fact that for all z, if z were a carpenter, then z is a politician, and that for all z, if z were a politician, then z is a carpenter we can infer that for all z, z is a carpenter if and only if z is a politician via universal biconditional introduction. From the fact that for all z, z is a politician if and only if z is a carpenter we can infer that for all z, if z were a politician, then z is a carpenter via universal biconditional elimination. From the fact that for all z, z is a carpenter if and only if z is a politician we can infer that for all z, if z were a carpenter, then z is a politician via universal biconditional elimination. From the fact that for all z, if z were a politician, then z is a carpenter, and that for all z, if z were a carpenter, then z is a politician we can infer that for all z, z is a politician if and only if z is a carpenter via universal biconditional introduction. Finally, from the fact that for all z, z is a politician if and only if z is a carpenter we can infer that for all z, if z were a carpenter, then z is a politician via universal biconditional elimination, which contradicts that There is at least one z for which the fact that z were a carpenter does not imply that z is a politician. Therefore, the answer is no.