Consider the following premises. For all x_6, if x_6 were a janitor, then x_6 is reading a book. For all x_6, x_6 is a janitor or x_6 will listen to a song. For all x_6, if x_6 listens to a song, then x_6 is writing a letter. For all x_6, x_6 will write a letter or x_6 will not read a book. Can we infer the following from them? If we can, name the inference rule being used: There is at least one x_6 for which x_6 will not write a letter.
 From the fact that for all x_6, x_6 will write a letter or x_6 will not read a book we can infer that for all x_6, if x_6 reads a book, then x_6 is writing a letter via universal rewriting. From the fact that for all x_6, if x_6 were a janitor, then x_6 is reading a book, and that for all x_6, if x_6 reads a book, then x_6 is writing a letter we can infer that for all x_6, if x_6 were a janitor, then x_6 is writing a letter via universal transitivity. Finally, from the fact that for all x_6, if x_6 were a janitor, then x_6 is writing a letter, that for all x_6, if x_6 listens to a song, then x_6 is writing a letter, and that for all x_6, x_6 is a janitor or x_6 will listen to a song we can infer that for all x_6, x_6 will write a letter via universal disjunction elimination, which contradicts that There is at least one x_6 for which x_6 will not write a letter. Therefore, the answer is no.