Consider the following premises. There is at least one z for which if z were a bartender, then z is a writer. For all z, if z were a writer, then z is an artist. For all z, if z were a writer, then z is a bartender. For all z, if z were an artist, then z is a writer. Can we infer the following from them? If we can, name the inference rule being used: There is at least one z for which if z were an artist, then z is a bartender.
 From the fact that for all z, if z were a writer, then z is an artist, and that for all z, if z were an artist, then z is a writer we can infer that for all z, z is a writer if and only if z is an artist via universal biconditional introduction. From the fact that for all z, if z were a writer, then z is a bartender, and that there is at least one z for which if z were a bartender, then z is a writer we can infer that there is at least one z for which z is a writer if and only if z is a bartender via existential biconditional introduction. From the fact that there is at least one z for which z is a writer if and only if z is a bartender we can infer that there is at least one z for which if z were a writer, then z is a bartender via existential biconditional elimination. From the fact that for all z, z is a writer if and only if z is an artist we can infer that for all z, if z were an artist, then z is a writer via universal biconditional elimination. Finally, from the fact that for all z, if z were an artist, then z is a writer, and that there is at least one z for which if z were a writer, then z is a bartender we can infer that there is at least one z for which if z were an artist, then z is a bartender via existential transitivity. Therefore, the answer is yes.