Consider the following premises. For all x, x will play tennis or x will not work. There is at least one x for which x will play tennis if and only if x will work. Can we infer the following from them? There is at least one x for which x will work if and only if x will play tennis.
 From the fact that for all x, x will play tennis or x will not work we can infer that for all x, if x works, then x is playing tennis. From the fact that there is at least one x for which x will play tennis if and only if x will work we can infer that there is at least one x for which if x plays tennis, then x is working. Finally, from the fact that for all x, if x works, then x is playing tennis, and that there is at least one x for which if x plays tennis, then x is working we can infer that there is at least one x for which x will work if and only if x will play tennis. Therefore, the answer is yes.