Consider the following premises. For all x_8, x_8 will sleep if and only if x_8 will play a game. There is at least one x_8 for which if x_8 sleeps, then x_8 is playing squash. For all x_8, if x_8 plays squash, then x_8 is playing a game. Can we infer the following from them? If we can, name the inference rule being used: There is at least one x_8 for which x_8 will play squash if and only if x_8 will sleep.
 From the fact that for all x_8, x_8 will sleep if and only if x_8 will play a game we can infer that for all x_8, if x_8 plays a game, then x_8 is sleeping via universal biconditional elimination. From the fact that for all x_8, if x_8 plays squash, then x_8 is playing a game, and that for all x_8, if x_8 plays a game, then x_8 is sleeping we can infer that for all x_8, if x_8 plays squash, then x_8 is sleeping via universal transitivity. Finally, from the fact that for all x_8, if x_8 plays squash, then x_8 is sleeping, and that there is at least one x_8 for which if x_8 sleeps, then x_8 is playing squash we can infer that there is at least one x_8 for which x_8 will play squash if and only if x_8 will sleep via existential biconditional introduction. Therefore, the answer is yes.