Consider the following premises. For all x_2, x_2 is a mechanic or x_2 will play tennis. There is at least one x_2 for which x_2 is excited if and only if x_2 will play tennis. For all x_2, x_2 is a mechanic if and only if x_2 is excited. Can we infer the following from them? If we can, name the inference rule being used: There is at least one x_2 for which x_2 is excited.
 From the fact that there is at least one x_2 for which x_2 is excited if and only if x_2 will play tennis we can infer that there is at least one x_2 for which if x_2 plays tennis, then x_2 is excited via existential biconditional elimination. From the fact that for all x_2, x_2 is a mechanic if and only if x_2 is excited we can infer that for all x_2, if x_2 were a mechanic, then x_2 is excited via universal biconditional elimination. Finally, from the fact that for all x_2, if x_2 were a mechanic, then x_2 is excited, that there is at least one x_2 for which if x_2 plays tennis, then x_2 is excited, and that for all x_2, x_2 is a mechanic or x_2 will play tennis we can infer that there is at least one x_2 for which x_2 is excited via existential disjunction elimination. Therefore, the answer is yes.