Consider the following premises. For all z_1, if z_1 plays squash, then z_1 is sleeping. For all z_1, z_1 is not a bartender. For all z_1, if z_1 were a politician, then z_1 is a bartender. For all z_1, z_1 will sleep if and only if z_1 is a politician. There is at least one z_1 for which if z_1 sleeps, then z_1 is playing squash. 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 z_1 will not play squash.
 From the fact that for all z_1, if z_1 plays squash, then z_1 is sleeping, and that there is at least one z_1 for which if z_1 sleeps, then z_1 is playing squash we can infer that there is at least one z_1 for which z_1 will play squash if and only if z_1 will sleep via existential biconditional introduction. From the fact that for all z_1, if z_1 were a politician, then z_1 is a bartender, and that for all z_1, z_1 is not a bartender we can infer that for all z_1, z_1 is not a politician via universal modus tollens. From the fact that for all z_1, z_1 will sleep if and only if z_1 is a politician, and that for all z_1, z_1 is not a politician we can infer that for all z_1, z_1 will not sleep via universal biconditional elimination. Finally, from the fact that there is at least one z_1 for which z_1 will play squash if and only if z_1 will sleep, and that for all z_1, z_1 will not sleep we can infer that there is at least one z_1 for which z_1 will not play squash via existential biconditional elimination. Therefore, the answer is yes.