Consider the following premises. For all z, z is a nurse if and only if z will go running. For all z, z will go running if and only if z will play a game. There is at least one z for which z will play a game if and only if z will take a plane. For all z, z is not a nurse. Can we infer the following from them? If we can, name the inference rule being used: For all z, z will take a plane.
 From the fact that for all z, z is a nurse if and only if z will go running, and that for all z, z is not a nurse we can infer that for all z, z will not go running via universal biconditional elimination. From the fact that for all z, z will go running if and only if z will play a game, and that for all z, z will not go running we can infer that for all z, z will not play a game via universal biconditional elimination. Finally, from the fact that there is at least one z for which z will play a game if and only if z will take a plane, and that for all z, z will not play a game we can infer that there is at least one z for which z will not take a plane via existential biconditional elimination, which contradicts that For all z, z will take a plane. Therefore, the answer is no.