Consider the following premises. For all x_8, if x_8 climbs a mountain, then x_8 is bored. For all x_8, if x_8 were bored, then x_8 is climbing a mountain. There is at least one x_8 for which x_8 will climb a mountain if and only if x_8 will cook. For all x_8, x_8 will not cook. Can we infer the following from them? If we can, name the inference rule being used: For all x_8, x_8 is bored.
 From the fact that there is at least one x_8 for which x_8 will climb a mountain if and only if x_8 will cook, and that for all x_8, x_8 will not cook we can infer that there is at least one x_8 for which x_8 will not climb a mountain via existential biconditional elimination. From the fact that for all x_8, if x_8 were bored, then x_8 is climbing a mountain, and that for all x_8, if x_8 climbs a mountain, then x_8 is bored we can infer that for all x_8, x_8 is bored if and only if x_8 will climb a mountain via universal biconditional introduction. Finally, from the fact that for all x_8, x_8 is bored if and only if x_8 will climb a mountain, and that there is at least one x_8 for which x_8 will not climb a mountain we can infer that there is at least one x_8 for which x_8 is not bored via existential biconditional elimination, which contradicts that For all x_8, x_8 is bored. Therefore, the answer is no.