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