Consider the following premises. For all x_8, x_8 is a politician or x_8 will sleep. For all x_8, if x_8 makes tea, then x_8 is climbing a mountain. For all x_8, if x_8 were a politician, then x_8 is making tea. There is at least one x_8 for which x_8 will sleep if and only if x_8 will climb a mountain. Can we infer the following from them? If we can, name the inference rule being used: There is at least one x_8 for which x_8 will climb a mountain.
 From the fact that there is at least one x_8 for which x_8 will sleep if and only if x_8 will climb a mountain we can infer that there is at least one x_8 for which if x_8 sleeps, then x_8 is climbing a mountain via existential biconditional elimination. From the fact that for all x_8, if x_8 were a politician, then x_8 is making tea, and that for all x_8, if x_8 makes tea, then x_8 is climbing a mountain we can infer that for all x_8, if x_8 were a politician, then x_8 is climbing a mountain via universal transitivity. Finally, from the fact that for all x_8, if x_8 were a politician, then x_8 is climbing a mountain, that there is at least one x_8 for which if x_8 sleeps, then x_8 is climbing a mountain, and that for all x_8, x_8 is a politician or x_8 will sleep we can infer that there is at least one x_8 for which x_8 will climb a mountain via existential disjunction elimination. Therefore, the answer is yes.