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