Consider the following premises. For all x_11, x_11 will not work. For all x_11, x_11 is a scientist if and only if x_11 is an astronaut. There is at least one x_11 for which x_11 will play a game if and only if x_11 is a scientist. For all x_11, x_11 will work if and only if x_11 is an astronaut. Can we infer the following from them? If we can, name the inference rule being used: There is at least one x_11 for which x_11 will not play a game.
 From the fact that for all x_11, x_11 is a scientist if and only if x_11 is an astronaut we can infer that for all x_11, if x_11 were a scientist, then x_11 is an astronaut via universal biconditional elimination. From the fact that for all x_11, x_11 will work if and only if x_11 is an astronaut, and that for all x_11, x_11 will not work we can infer that for all x_11, x_11 is not an astronaut via universal biconditional elimination. From the fact that for all x_11, if x_11 were a scientist, then x_11 is an astronaut, and that for all x_11, x_11 is not an astronaut we can infer that for all x_11, x_11 is not a scientist via universal modus tollens. Finally, from the fact that there is at least one x_11 for which x_11 will play a game if and only if x_11 is a scientist, and that for all x_11, x_11 is not a scientist we can infer that there is at least one x_11 for which x_11 will not play a game via existential biconditional elimination. Therefore, the answer is yes.