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