Translate the following inference to logic notation: For all x, if x_5 writes a letter, then x_5 is listening to a song. There is at least one x for which if x_5 writes a letter, then x_5 is listening to a song. For all x, if x_5 listens to a song, then x_5 is writing a letter. Therefore there is at least one x for which x_5 will write a letter if and only if x_5 will listen to a song.
forall x: P(x) -> Q(x). exists x: P(x) -> Q(x). forall x: Q(x) -> P(x). Therefore exists x: P(x) <-> Q(x).