Consider the following premises. For all x_7, if x_7 sleeps, then x_7 is playing a game. For all x_7, x_7 will play a game if and only if x_7 will work. For all x_7, if x_7 works, then x_7 is sleeping. Can we infer the following from them? If we can, name the inference rule being used: There is at least one x_7 for which if is not true that x_7 will work if and only if x_7 will sleep.
 From the fact that for all x_7, x_7 will play a game if and only if x_7 will work we can infer that for all x_7, if x_7 plays a game, then x_7 is working via universal biconditional elimination. From the fact that for all x_7, if x_7 sleeps, then x_7 is playing a game, and that for all x_7, if x_7 plays a game, then x_7 is working we can infer that for all x_7, if x_7 sleeps, then x_7 is working via universal transitivity. Finally, from the fact that for all x_7, if x_7 works, then x_7 is sleeping, and that for all x_7, if x_7 sleeps, then x_7 is working we can infer that for all x_7, x_7 will work if and only if x_7 will sleep via universal biconditional introduction, which contradicts that There is at least one x_7 for which if is not true that x_7 will work if and only if x_7 will sleep. Therefore, the answer is no.