Consider the following premises. For all z, if z plays tennis, then z is playing squash. For all z, if z plays squash, then z is a mechanic. For all z, z is not a mechanic. For all z, z will play tennis or z will not play squash. Can we infer the following from them? For all z, z will not play tennis.
 From the fact that for all z, if z plays squash, then z is a mechanic, and that for all z, z is not a mechanic we can infer that for all z, z will not play squash. From the fact that for all z, z will play tennis or z will not play squash we can infer that for all z, if z plays squash, then z is playing tennis. From the fact that for all z, if z plays tennis, then z is playing squash, and that for all z, if z plays squash, then z is playing tennis we can infer that for all z, z will play tennis if and only if z will play squash. Finally, from the fact that for all z, z will play tennis if and only if z will play squash, and that for all z, z will not play squash we can infer that for all z, z will not play tennis. Therefore, the answer is yes.