Consider the following premises. For all x_6, if x_6 were a janitor, then x_6 is playing a game. There is at least one x_6 for which if x_6 were a janitor, then x_6 is making tea. For all x_6, if x_6 makes tea, then x_6 is a janitor. Can we infer the following from them? If we can, name the inference rule being used: There is at least one x_6 for which if x_6 makes tea, then x_6 is playing a game.
 From the fact that there is at least one x_6 for which if x_6 were a janitor, then x_6 is making tea, and that for all x_6, if x_6 makes tea, then x_6 is a janitor we can infer that there is at least one x_6 for which x_6 is a janitor if and only if x_6 will make tea via existential biconditional introduction. From the fact that there is at least one x_6 for which x_6 is a janitor if and only if x_6 will make tea we can infer that there is at least one x_6 for which if x_6 makes tea, then x_6 is a janitor via existential biconditional elimination. Finally, from the fact that there is at least one x_6 for which if x_6 makes tea, 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 there is at least one x_6 for which if x_6 makes tea, then x_6 is playing a game via existential transitivity. Therefore, the answer is yes.