Consider the following premises. John is not playing a game. For all x_6, if x_6 were a janitor, then x_6 is playing a game. For all x_6, x_6 will play a game if and only if x_6 is a janitor. Can we infer the following from them? If we can, name the inference rule being used: John is a janitor.
 From the fact that for all x_6, x_6 will play a game if and only if x_6 is a janitor we can infer that for all x_6, if x_6 plays a game, then x_6 is a janitor via universal biconditional elimination. From the fact that for all x_6, if x_6 plays a game, then x_6 is a janitor, and that for all x_6, if x_6 were a janitor, then x_6 is playing a game we can infer that for all x_6, x_6 will play a game if and only if x_6 is a janitor via universal biconditional introduction. From the fact that for all x_6, x_6 will play a game if and only if x_6 is a janitor we can infer that John is playing a game if and only if John is a janitor via universal instantiation. Finally, from the fact that John is playing a game if and only if John is a janitor, and that John is not playing a game we can infer that John is not a janitor via biconditional elimination, which contradicts that John is a janitor. Therefore, the answer is no.