Consider the following premises. For all z_1, if z_1 cooks, then z_1 is climbing a mountain. For all z_1, z_1 is a mechanic or z_1 will not climb a mountain. For all z_1, z_1 will climb a mountain if and only if z_1 will cook. Can we infer the following from them? There is at least one z_1 for which the fact that z_1 cooks does not imply that z_1 is a mechanic.
 From the fact that for all z_1, z_1 will climb a mountain if and only if z_1 will cook we can infer that for all z_1, if z_1 climbs a mountain, then z_1 is cooking. From the fact that for all z_1, z_1 is a mechanic or z_1 will not climb a mountain we can infer that for all z_1, if z_1 climbs a mountain, then z_1 is a mechanic. From the fact that for all z_1, if z_1 cooks, then z_1 is climbing a mountain, and that for all z_1, if z_1 climbs a mountain, then z_1 is cooking we can infer that for all z_1, z_1 will cook if and only if z_1 will climb a mountain. From the fact that for all z_1, z_1 will cook if and only if z_1 will climb a mountain we can infer that for all z_1, if z_1 cooks, then z_1 is climbing a mountain. Finally, from the fact that for all z_1, if z_1 cooks, then z_1 is climbing a mountain, and that for all z_1, if z_1 climbs a mountain, then z_1 is a mechanic we can infer that for all z_1, if z_1 cooks, then z_1 is a mechanic, which contradicts that There is at least one z_1 for which the fact that z_1 cooks does not imply that z_1 is a mechanic. Therefore, the answer is no.