Consider the following premises. For all x_3, x_3 will not write a letter or x_3 will climb a mountain. There is at least one x_3 for which if x_3 works, then x_3 is excited. For all x_3, x_3 will write a letter. For all x_3, if x_3 climbs a mountain, then x_3 is excited. Can we infer the following from them? If we can, name the inference rule being used: For all x_3, x_3 is not excited.
 From the fact that for all x_3, x_3 will write a letter we can infer that for all x_3, x_3 will write a letter or x_3 will work via universal addition. From the fact that for all x_3, x_3 will write a letter or x_3 will work, and that for all x_3, x_3 will not write a letter or x_3 will climb a mountain we can infer that for all x_3, x_3 will work or x_3 will climb a mountain via universal resolution. Finally, from the fact that there is at least one x_3 for which if x_3 works, then x_3 is excited, that for all x_3, if x_3 climbs a mountain, then x_3 is excited, and that for all x_3, x_3 will work or x_3 will climb a mountain we can infer that there is at least one x_3 for which x_3 is excited via existential disjunction elimination, which contradicts that For all x_3, x_3 is not excited. Therefore, the answer is no.