Consider the following premises. For all x_9, if x_9 were sad, then x_9 is playing a game. For all x_8, x_8 will play a game. For all x_9, if x_9 plays a game, then x_9 is sad. Can we infer the following from them? If we can, name the inference rule being used: Karen is not sad.
 From the fact that for all x_9, if x_9 plays a game, then x_9 is sad, and that for all x_9, if x_9 were sad, then x_9 is playing a game we can infer that for all x_9, x_9 will play a game if and only if x_9 is sad via universal biconditional introduction. From the fact that for all x_8, x_8 will play a game we can infer that Karen is playing a game via universal instantiation. From the fact that for all x_9, x_9 will play a game if and only if x_9 is sad we can infer that Karen is playing a game if and only if Karen is sad via universal instantiation. From the fact that Karen is playing a game if and only if Karen is sad we can infer that if Karen plays a game, then Karen is sad via biconditional elimination. Finally, from the fact that if Karen plays a game, then Karen is sad, and that Karen is playing a game we can infer that Karen is sad via modus ponens, which contradicts that Karen is not sad. Therefore, the answer is no.