Consider the following premises. There is at least one z for which z is a lawyer or z is not a police officer. For all z, if z were a lawyer, then z is sad. For all z, z is a police officer if and only if z is sad. 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 sad if and only if z is a police officer.
 From the fact that there is at least one z for which z is a lawyer or z is not a police officer we can infer that there is at least one z for which if z were a police officer, then z is a lawyer via existential rewriting. From the fact that there is at least one z for which if z were a police officer, then z is a lawyer, and that for all z, if z were a lawyer, then z is sad we can infer that there is at least one z for which if z were a police officer, then z is sad via existential transitivity. From the fact that for all z, z is a police officer if and only if z is sad we can infer that for all z, if z were sad, then z is a police officer via universal biconditional elimination. Finally, from the fact that for all z, if z were sad, then z is a police officer, and that there is at least one z for which if z were a police officer, then z is sad we can infer that there is at least one z for which z is sad if and only if z is a police officer via existential biconditional introduction. Therefore, the answer is yes.