Consider the following premises. For all z, z is not a politician. For all z, if z were a lawyer, then z is cooking. For all z, if z listens to a song, then z is a lawyer. There is at least one z for which if z cooks, then z is listening to a song. For all z, z is a politician if and only if z is a lawyer. 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 will not cook.
 From the fact that for all z, z is a politician if and only if z is a lawyer, and that for all z, z is not a politician we can infer that for all z, z is not a lawyer via universal biconditional elimination. From the fact that there is at least one z for which if z cooks, then z is listening to a song, and that for all z, if z listens to a song, then z is a lawyer we can infer that there is at least one z for which if z cooks, then z is a lawyer via existential transitivity. From the fact that for all z, if z were a lawyer, then z is cooking, and that there is at least one z for which if z cooks, then z is a lawyer we can infer that there is at least one z for which z is a lawyer if and only if z will cook via existential biconditional introduction. Finally, from the fact that there is at least one z for which z is a lawyer if and only if z will cook, and that for all z, z is not a lawyer we can infer that there is at least one z for which z will not cook via existential biconditional elimination. Therefore, the answer is yes.