Consider the following premises. For all x_14, if x_14 were an astronaut, then x_14 is sleeping. For all x_14, if x_14 reads a book, then x_14 is an astronaut. For all x_14, x_14 will read a book or x_14 will play a game. There is at least one x_14 for which x_14 is an astronaut if and only if x_14 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 x_14 for which x_14 will sleep.
 From the fact that there is at least one x_14 for which x_14 is an astronaut if and only if x_14 will play a game we can infer that there is at least one x_14 for which if x_14 plays a game, then x_14 is an astronaut via existential biconditional elimination. From the fact that for all x_14, if x_14 reads a book, then x_14 is an astronaut, that there is at least one x_14 for which if x_14 plays a game, then x_14 is an astronaut, and that for all x_14, x_14 will read a book or x_14 will play a game we can infer that there is at least one x_14 for which x_14 is an astronaut via existential disjunction elimination. Finally, from the fact that for all x_14, if x_14 were an astronaut, then x_14 is sleeping, and that there is at least one x_14 for which x_14 is an astronaut we can infer that there is at least one x_14 for which x_14 will sleep via existential modus ponens. Therefore, the answer is yes.