Consider the following premises. For all z, if z plays a game, then z is an office clerk. There is at least one z for which z will make tea if and only if z is an office clerk. For all z, if z makes tea, then z is an astronaut. For all z, if z were an astronaut, then z is playing a game. Can we infer the following from them? If we can, name the inference rule being used: For all z, if is not true that z will make tea if and only if z is an office clerk.
 From the fact that for all z, if z were an astronaut, then z is playing a game, and that for all z, if z plays a game, then z is an office clerk we can infer that for all z, if z were an astronaut, then z is an office clerk via universal transitivity. From the fact that there is at least one z for which z will make tea if and only if z is an office clerk we can infer that there is at least one z for which if z were an office clerk, then z is making tea via existential biconditional elimination. From the fact that for all z, if z makes tea, then z is an astronaut, and that for all z, if z were an astronaut, then z is an office clerk we can infer that for all z, if z makes tea, then z is an office clerk via universal transitivity. Finally, from the fact that for all z, if z makes tea, then z is an office clerk, and that there is at least one z for which if z were an office clerk, then z is making tea we can infer that there is at least one z for which z will make tea if and only if z is an office clerk via existential biconditional introduction, which contradicts that For all z, if is not true that z will make tea if and only if z is an office clerk. Therefore, the answer is no.