Consider the following premises. There is at least one x_5 for which x_5 will work. For all x_5, x_5 will not write a letter. For all x_5, x_5 is a carpenter or x_5 is a doctor. For all x_5, if x_5 were a carpenter, then x_5 is writing a letter. Can we infer the following from them? If we can, name the inference rule being used: There is at least one x_5 for which x_5 will work and x_5 is a doctor.
 From the fact that for all x_5, if x_5 were a carpenter, then x_5 is writing a letter, and that for all x_5, x_5 will not write a letter we can infer that for all x_5, x_5 is not a carpenter via universal modus tollens. From the fact that for all x_5, x_5 is a carpenter or x_5 is a doctor, and that for all x_5, x_5 is not a carpenter we can infer that for all x_5, x_5 is a doctor via universal disjunctive syllogism. Finally, from the fact that there is at least one x_5 for which x_5 will work, and that for all x_5, x_5 is a doctor we can infer that there is at least one x_5 for which x_5 will work and x_5 is a doctor via existential conjunction. Therefore, the answer is yes.