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