Consider the following premises. There is at least one x_10 for which if x_10 were a carpenter, then x_10 is a doctor. For all x_10, x_10 will read a book. For all x_10, if x_10 were a doctor, then x_10 is writing a letter. For all x_10, if x_10 reads a book, then x_10 is a carpenter. Can we infer the following from them? If we can, name the inference rule being used: There is at least one x_10 for which x_10 will write a letter.
 From the fact that for all x_10, if x_10 reads a book, then x_10 is a carpenter, and that for all x_10, x_10 will read a book we can infer that for all x_10, x_10 is a carpenter via universal modus ponens. From the fact that there is at least one x_10 for which if x_10 were a carpenter, then x_10 is a doctor, and that for all x_10, x_10 is a carpenter we can infer that there is at least one x_10 for which x_10 is a doctor via existential modus ponens. Finally, from the fact that for all x_10, if x_10 were a doctor, then x_10 is writing a letter, and that there is at least one x_10 for which x_10 is a doctor we can infer that there is at least one x_10 for which x_10 will write a letter via existential modus ponens. Therefore, the answer is yes.