Consider the following premises. If John climbs a mountain, then Richard is taking a plane. For all x_10, if x_10 plays a game, then x_10 is climbing a mountain. If Richard takes a plane, then John is playing a game. Can we infer the following from them? If we can, name the inference rule being used: If is not true that John is climbing a mountain if and only if Richard is taking a plane.
 From the fact that for all x_10, if x_10 plays a game, then x_10 is climbing a mountain we can infer that if John plays a game, then John is climbing a mountain via universal instantiation. From the fact that if Richard takes a plane, then John is playing a game, and that if John plays a game, then John is climbing a mountain we can infer that if Richard takes a plane, then John is climbing a mountain via transitivity. Finally, from the fact that if John climbs a mountain, then Richard is taking a plane, and that if Richard takes a plane, then John is climbing a mountain we can infer that John is climbing a mountain if and only if Richard is taking a plane via biconditional introduction, which contradicts that If is not true that John is climbing a mountain if and only if Richard is taking a plane. Therefore, the answer is no.