Consider the following premises. For all y, y will go running or y will not play tennis. There is at least one y for which if y goes running, then y is playing tennis. For all y, if y goes running, then y is playing tennis. Can we infer the following from them? For all y, if is not true that y will go running if and only if y will play tennis.
 From the fact that for all y, y will go running or y will not play tennis we can infer that for all y, if y plays tennis, then y is running. Finally, from the fact that for all y, if y goes running, then y is playing tennis, and that for all y, if y plays tennis, then y is running we can infer that for all y, y will go running if and only if y will play tennis, which contradicts that For all y, if is not true that y will go running if and only if y will play tennis. Therefore, the answer is no.