Consider the following premises. For all x_8, x_8 will not listen to a song. For all x_8, x_8 will drink water if and only if x_8 will listen to a song. For all x_8, x_8 will drink water if and only if x_8 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_8 for which x_8 will write a letter.
 From the fact that for all x_8, x_8 will drink water if and only if x_8 will write a letter we can infer that for all x_8, if x_8 writes a letter, then x_8 is drinking water via universal biconditional elimination. From the fact that for all x_8, x_8 will drink water if and only if x_8 will listen to a song, and that for all x_8, x_8 will not listen to a song we can infer that for all x_8, x_8 will not drink water via universal biconditional elimination. Finally, from the fact that for all x_8, if x_8 writes a letter, then x_8 is drinking water, and that for all x_8, x_8 will not drink water we can infer that for all x_8, x_8 will not write a letter via universal modus tollens, which contradicts that There is at least one x_8 for which x_8 will write a letter. Therefore, the answer is no.