Consider the following premises. For all x_6, if x_6 were bored, then x_6 is a scientist. For all x_6, if x_6 plays a game, then x_6 is bored. There is at least one x_6 for which if x_6 were a scientist, then x_6 is a politician. For all x_6, if x_6 were a politician, then x_6 is playing a game. Can we infer the following from them? There is at least one x_6 for which x_6 is bored if and only if x_6 is a politician.
 From the fact that for all x_6, if x_6 were bored, then x_6 is a scientist, and that there is at least one x_6 for which if x_6 were a scientist, then x_6 is a politician we can infer that there is at least one x_6 for which if x_6 were bored, then x_6 is a politician. From the fact that for all x_6, if x_6 were a politician, then x_6 is playing a game, and that for all x_6, if x_6 plays a game, then x_6 is bored we can infer that for all x_6, if x_6 were a politician, then x_6 is bored. Finally, from the fact that there is at least one x_6 for which if x_6 were bored, then x_6 is a politician, and that for all x_6, if x_6 were a politician, then x_6 is bored we can infer that there is at least one x_6 for which x_6 is bored if and only if x_6 is a politician. Therefore, the answer is yes.