Consider the following premises. For all x_14, x_14 will listen to a song if and only if x_14 will climb a mountain. There is at least one x_14 for which if x_14 climbs a mountain, then x_14 is listening to a song. For all x_14, x_14 will listen to a song if and only if x_14 will write a letter. Can we infer the following from them? If we can, name the inference rule being used: There is at least one x_14 for which if x_14 climbs a mountain, then x_14 is writing a letter.
 From the fact that for all x_14, x_14 will listen to a song if and only if x_14 will climb a mountain we can infer that for all x_14, if x_14 listens to a song, then x_14 is climbing a mountain via universal biconditional elimination. From the fact that for all x_14, if x_14 listens to a song, then x_14 is climbing a mountain, and that there is at least one x_14 for which if x_14 climbs a mountain, then x_14 is listening to a song we can infer that there is at least one x_14 for which x_14 will listen to a song if and only if x_14 will climb a mountain via existential biconditional introduction. From the fact that for all x_14, x_14 will listen to a song if and only if x_14 will write a letter we can infer that for all x_14, if x_14 listens to a song, then x_14 is writing a letter via universal biconditional elimination. From the fact that there is at least one x_14 for which x_14 will listen to a song if and only if x_14 will climb a mountain we can infer that there is at least one x_14 for which if x_14 climbs a mountain, then x_14 is listening to a song via existential biconditional elimination. Finally, from the fact that there is at least one x_14 for which if x_14 climbs a mountain, then x_14 is listening to a song, and that for all x_14, if x_14 listens to a song, then x_14 is writing a letter we can infer that there is at least one x_14 for which if x_14 climbs a mountain, then x_14 is writing a letter via existential transitivity. Therefore, the answer is yes.