Consider the following premises. For all x_9, x_9 will not write a letter. There is at least one x_9 for which if x_9 climbs a mountain, then x_9 is listening to a song. For all x_9, x_9 will write a letter if and only if x_9 will listen to a song. Can we infer the following from them? If we can, name the inference rule being used: There is at least one x_9 for which x_9 will not climb a mountain.
 From the fact that for all x_9, x_9 will write a letter if and only if x_9 will listen to a song we can infer that for all x_9, if x_9 listens to a song, then x_9 is writing a letter via universal biconditional elimination. From the fact that there is at least one x_9 for which if x_9 climbs a mountain, then x_9 is listening to a song, and that for all x_9, if x_9 listens to a song, then x_9 is writing a letter we can infer that there is at least one x_9 for which if x_9 climbs a mountain, then x_9 is writing a letter via existential transitivity. Finally, from the fact that there is at least one x_9 for which if x_9 climbs a mountain, then x_9 is writing a letter, and that for all x_9, x_9 will not write a letter we can infer that there is at least one x_9 for which x_9 will not climb a mountain via existential modus tollens. Therefore, the answer is yes.