Consider the following premises. There is at least one z_4 for which if z_4 were an artist, then z_4 is listening to a song. For all z_4, z_4 will listen to a song if and only if z_4 is a nurse. For all z_4, if z_4 were a nurse, then z_4 is an artist. Can we infer the following from them? If we can, name the inference rule being used: There is at least one z_4 for which z_4 is a nurse if and only if z_4 is an artist.
 From the fact that for all z_4, z_4 will listen to a song if and only if z_4 is a nurse we can infer that for all z_4, if z_4 listens to a song, then z_4 is a nurse via universal biconditional elimination. From the fact that there is at least one z_4 for which if z_4 were an artist, then z_4 is listening to a song, and that for all z_4, if z_4 listens to a song, then z_4 is a nurse we can infer that there is at least one z_4 for which if z_4 were an artist, then z_4 is a nurse via existential transitivity. Finally, from the fact that for all z_4, if z_4 were a nurse, then z_4 is an artist, and that there is at least one z_4 for which if z_4 were an artist, then z_4 is a nurse we can infer that there is at least one z_4 for which z_4 is a nurse if and only if z_4 is an artist via existential biconditional introduction. Therefore, the answer is yes.