Consider the following premises. If it rains, then Mary is playing a game. Mary is not playing a game. It is raining or Mary is not playing a game. Can we infer the following from them? It is raining.
 From the fact that it is raining or Mary is not playing a game we can infer that if Mary plays a game, then it is raining. From the fact that if it rains, then Mary is playing a game, and that if Mary plays a game, then it is raining we can infer that it is raining if and only if Mary is playing a game. Finally, from the fact that it is raining if and only if Mary is playing a game, and that Mary is not playing a game we can infer that it is not raining, which contradicts that It is raining. Therefore, the answer is no.