Consider the following premises. For all z, if z plays a game, then z is an office clerk. For all z, if z were an astronaut, then z is not an office clerk. For all z, z will not play squash. There is at least one z for which z will play squash or z will play a game. 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 an astronaut.
 From the fact that there is at least one z for which z will play squash or z will play a game, and that for all z, z will not play squash we can infer that there is at least one z for which z will play a game via existential disjunctive syllogism. From the fact that for all z, if z plays a game, then z is an office clerk, and that there is at least one z for which z will play a game we can infer that there is at least one z for which z is an office clerk via existential modus ponens. Finally, from the fact that for all z, if z were an astronaut, then z is not an office clerk, and that there is at least one z for which z is an office clerk we can infer that there is at least one z for which z is not an astronaut via existential modus tollens. Therefore, the answer is yes.