Consider the following premises. There is at least one x_5 for which if x_5 plays a game, then x_5 is bored. For all x_5, if x_5 were bored, then x_5 is fast. For all x_5, x_5 is fast or x_5 will play a game. For all x_5, if x_5 were fast, then x_5 is bored. Can we infer the following from them? If we can, name the inference rule being used: There is at least one x_5 for which x_5 is bored.
 From the fact that for all x_5, if x_5 were bored, then x_5 is fast, and that for all x_5, if x_5 were fast, then x_5 is bored we can infer that for all x_5, x_5 is bored if and only if x_5 is fast via universal biconditional introduction. From the fact that for all x_5, x_5 is bored if and only if x_5 is fast we can infer that for all x_5, if x_5 were fast, then x_5 is bored via universal biconditional elimination. Finally, from the fact that for all x_5, if x_5 were fast, then x_5 is bored, that there is at least one x_5 for which if x_5 plays a game, then x_5 is bored, and that for all x_5, x_5 is fast or x_5 will play a game we can infer that there is at least one x_5 for which x_5 is bored via existential disjunction elimination. Therefore, the answer is yes.