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