Consider the following premises. There is at least one y for which y will listen to a song if and only if y is a scientist. For all y, y is an astronaut if and only if y is a scientist. For all y, y will not listen to a song. Can we infer the following from them? If we can, name the inference rule being used: For all y, y is an astronaut.
 From the fact that for all y, y is an astronaut if and only if y is a scientist we can infer that for all y, if y were an astronaut, then y is a scientist via universal biconditional elimination. From the fact that there is at least one y for which y will listen to a song if and only if y is a scientist, and that for all y, y will not listen to a song we can infer that there is at least one y for which y is not a scientist via existential biconditional elimination. Finally, from the fact that for all y, if y were an astronaut, then y is a scientist, and that there is at least one y for which y is not a scientist we can infer that there is at least one y for which y is not an astronaut via existential modus tollens, which contradicts that For all y, y is an astronaut. Therefore, the answer is no.