Consider the following premises. For all x, x will listen to a song or x will not climb a mountain. There is at least one x for which x will not climb a mountain. For all x, if x listens to a song, then x is climbing a mountain. Can we infer the following from them? If we can, name the inference rule being used: For all x, x will listen to a song.
 From the fact that for all x, x will listen to a song or x will not climb a mountain we can infer that for all x, if x climbs a mountain, then x is listening to a song via universal rewriting. From the fact that for all x, if x climbs a mountain, then x is listening to a song, and that for all x, if x listens to a song, then x is climbing a mountain we can infer that for all x, x will climb a mountain if and only if x will listen to a song via universal biconditional introduction. Finally, from the fact that for all x, x will climb a mountain if and only if x will listen to a song, and that there is at least one x for which x will not climb a mountain we can infer that there is at least one x for which x will not listen to a song via existential biconditional elimination, which contradicts that For all x, x will listen to a song. Therefore, the answer is no.