Consider the following premises. For all x_9, if x_9 plays a game, then x_9 is taking a plane. There is at least one x_9 for which x_9 is not an astronaut. For all x_9, if x_9 takes a plane, then x_9 is an astronaut. For all x_9, x_9 will play a game or x_9 is a carpenter. Can we infer the following from them? If we can, name the inference rule being used: There is at least one x_9 for which x_9 is a carpenter.
 From the fact that for all x_9, if x_9 plays a game, then x_9 is taking a plane, and that for all x_9, if x_9 takes a plane, then x_9 is an astronaut we can infer that for all x_9, if x_9 plays a game, then x_9 is an astronaut via universal transitivity. From the fact that for all x_9, if x_9 plays a game, then x_9 is an astronaut, and that there is at least one x_9 for which x_9 is not an astronaut we can infer that there is at least one x_9 for which x_9 will not play a game via existential modus tollens. Finally, from the fact that for all x_9, x_9 will play a game or x_9 is a carpenter, and that there is at least one x_9 for which x_9 will not play a game we can infer that there is at least one x_9 for which x_9 is a carpenter via existential disjunctive syllogism. Therefore, the answer is yes.