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