Consider the following premises. For all z, if z reads a book, then z is a mechanic. There is at least one z for which z is a mechanic if and only if z will play tennis. Can we infer the following from them? If we can, name the inference rule being used: There is at least one z for which if z reads a book, then z is playing tennis.
 From the fact that there is at least one z for which z is a mechanic if and only if z will play tennis we can infer that there is at least one z for which if z were a mechanic, then z is playing tennis via existential biconditional elimination. Finally, from the fact that for all z, if z reads a book, then z is a mechanic, and that there is at least one z for which if z were a mechanic, then z is playing tennis we can infer that there is at least one z for which if z reads a book, then z is playing tennis via existential transitivity. Therefore, the answer is yes.