Consider the following premises. For all z, if z plays tennis, then z is happy. There is at least one z for which if z were happy, then z is playing tennis. For all z, z will not play tennis. 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 not happy.
 From the fact that there is at least one z for which if z were happy, then z is playing tennis, and that for all z, if z plays tennis, then z is happy we can infer that there is at least one z for which z is happy if and only if z will play tennis via existential biconditional introduction. Finally, from the fact that there is at least one z for which z is happy if and only if z will play tennis, and that for all z, z will not play tennis we can infer that there is at least one z for which z is not happy via existential biconditional elimination. Therefore, the answer is yes.