Consider the following premises. For all z_3, if z_3 climbs a mountain, then z_3 is playing tennis. For all z_3, if z_3 plays tennis, then z_3 is climbing a mountain. There is at least one z_3 for which if z_3 plays tennis, then z_3 is climbing a mountain. Can we infer the following from them? For all z_3, if is not true that z_3 will play tennis if and only if z_3 will climb a mountain.
That contradicts the premises. Therefore the answer is no.