Consider the following premises. Michael is bored or it is not cloudy. If John plays a game, then it is cloudy. John is playing a game or Michael is not bored. Can we infer the following from them? John is playing a game if and only if Michael is bored.
 From the fact that Michael is bored or it is not cloudy we can infer that if it is cloudy, then Michael is bored. From the fact that if John plays a game, then it is cloudy, and that if it is cloudy, then Michael is bored we can infer that if John plays a game, then Michael is bored. From the fact that John is playing a game or Michael is not bored we can infer that if Michael were bored, then John is playing a game. Finally, from the fact that if John plays a game, then Michael is bored, and that if Michael were bored, then John is playing a game we can infer that John is playing a game if and only if Michael is bored. Therefore, the answer is yes.