Consider the following premises. For all x_10, x_10 is a mechanic or x_10 is a carpenter. There is at least one x_10 for which x_10 is not a mechanic. For all x_10, x_10 is sad. For all x_10, if x_10 were sad, then x_10 is drinking water. Can we infer the following from them? If we can, name the inference rule being used: For all x_10, either x_10 is a carpenter or x_10 will drink water are not true.
 From the fact that for all x_10, if x_10 were sad, then x_10 is drinking water, and that for all x_10, x_10 is sad we can infer that for all x_10, x_10 will drink water via universal modus ponens. From the fact that for all x_10, x_10 is a mechanic or x_10 is a carpenter, and that there is at least one x_10 for which x_10 is not a mechanic we can infer that there is at least one x_10 for which x_10 is a carpenter via existential disjunctive syllogism. Finally, from the fact that there is at least one x_10 for which x_10 is a carpenter, and that for all x_10, x_10 will drink water we can infer that some carpenters will drink water via existential conjunction, which contradicts that For all x_10, either x_10 is a carpenter or x_10 will drink water are not true. Therefore, the answer is no.