Consider the following premises. There is at least one z_1 for which z_1 will play tennis if and only if z_1 will sleep. For all z_1, if z_1 sleeps, then z_1 is playing tennis. 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 sleeps, then z_1 is playing tennis.
 From the fact that there is at least one z_1 for which z_1 will play tennis 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 tennis, then z_1 is sleeping via existential biconditional elimination. From the fact that there is at least one z_1 for which if z_1 plays tennis, then z_1 is sleeping, and that for all z_1, if z_1 sleeps, then z_1 is playing tennis we can infer that there is at least one z_1 for which z_1 will play tennis if and only if z_1 will sleep via existential biconditional introduction. Finally, from the fact that there is at least one z_1 for which z_1 will play tennis if and only if z_1 will sleep we can infer that there is at least one z_1 for which if z_1 sleeps, then z_1 is playing tennis via existential biconditional elimination. Therefore, the answer is yes.