Consider the following premises. There is at least one x for which if x makes tea, then x is playing a game. For all x, if x plays a game, then x is making tea. For all x, x will make tea or x will not play a game. Can we infer the following from them? If we can, name the inference rule being used: For all x, if is not true that x will make tea if and only if x will play a game.
 From the fact that there is at least one x for which if x makes tea, then x is playing a game, and that for all x, if x plays a game, then x is making tea we can infer that there is at least one x for which x will make tea if and only if x will play a game via existential biconditional introduction. From the fact that for all x, x will make tea or x will not play a game we can infer that for all x, if x plays a game, then x is making tea via universal rewriting. From the fact that there is at least one x for which x will make tea if and only if x will play a game we can infer that there is at least one x for which if x makes tea, then x is playing a game via existential biconditional elimination. Finally, from the fact that there is at least one x for which if x makes tea, then x is playing a game, and that for all x, if x plays a game, then x is making tea we can infer that there is at least one x for which x will make tea if and only if x will play a game via existential biconditional introduction, which contradicts that For all x, if is not true that x will make tea if and only if x will play a game. Therefore, the answer is no.