Consider the following premises. There is at least one x_12 for which x_12 is fast. For all x_12, if x_12 drinks water, then x_12 is writing a letter. For all x_12, if x_12 were fast, then x_12 is drinking water. Can we infer the following from them? If we can, name the inference rule being used: For all x_12, neither x_12 will write a letter nor x_12 will work.
 From the fact that for all x_12, if x_12 were fast, then x_12 is drinking water, and that there is at least one x_12 for which x_12 is fast we can infer that there is at least one x_12 for which x_12 will drink water via existential modus ponens. From the fact that for all x_12, if x_12 drinks water, then x_12 is writing a letter, and that there is at least one x_12 for which x_12 will drink water we can infer that there is at least one x_12 for which x_12 will write a letter via existential modus ponens. Finally, from the fact that there is at least one x_12 for which x_12 will write a letter we can infer that there is at least one x_12 for which x_12 will write a letter or x_12 will work via existential addition, which contradicts that For all x_12, neither x_12 will write a letter nor x_12 will work. Therefore, the answer is no.