Consider the following premises. For all x_9, if x_9 climbs a mountain, then x_9 is listening to a song. For all x_9, x_9 will climb a mountain if and only if x_9 is an astronaut. For all x_9, if x_9 listens to a song, then x_9 is climbing a mountain. 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 the fact that x_9 were an astronaut does not imply that x_9 is listening to a song.
 From the fact that for all x_9, if x_9 listens to a song, then x_9 is climbing a mountain, and that for all x_9, if x_9 climbs a mountain, then x_9 is listening to a song we can infer that for all x_9, x_9 will listen to a song if and only if x_9 will climb a mountain via universal biconditional introduction. From the fact that for all x_9, x_9 will listen to a song if and only if x_9 will climb a mountain we can infer that for all x_9, if x_9 climbs a mountain, then x_9 is listening to a song via universal biconditional elimination. From the fact that for all x_9, x_9 will climb a mountain if and only if x_9 is an astronaut we can infer that for all x_9, if x_9 were an astronaut, then x_9 is climbing a mountain via universal biconditional elimination. Finally, from the fact that for all x_9, if x_9 were an astronaut, then x_9 is climbing a mountain, and that for all x_9, if x_9 climbs a mountain, then x_9 is listening to a song we can infer that for all x_9, if x_9 were an astronaut, then x_9 is listening to a song via universal transitivity, which contradicts that There is at least one x_9 for which the fact that x_9 were an astronaut does not imply that x_9 is listening to a song. Therefore, the answer is no.