Consider the following premises. For all z_1, z_1 is an artist or z_1 will not sleep. There is at least one z_1 for which z_1 will play a game if and only if z_1 will sleep. Can we infer the following from them? If we can, name the inference rule being used: There is at least one z_1 for which if z_1 plays a game, then z_1 is an artist.
 From the fact that for all z_1, z_1 is an artist or z_1 will not sleep we can infer that for all z_1, if z_1 sleeps, then z_1 is an artist via universal rewriting. From the fact that there is at least one z_1 for which z_1 will play a game if and only if z_1 will sleep we can infer that there is at least one z_1 for which if z_1 plays a game, then z_1 is sleeping via existential biconditional elimination. Finally, from the fact that there is at least one z_1 for which if z_1 plays a game, then z_1 is sleeping, and that for all z_1, if z_1 sleeps, then z_1 is an artist we can infer that there is at least one z_1 for which if z_1 plays a game, then z_1 is an artist via existential transitivity. Therefore, the answer is yes.