Translate the following inference to logic notation: For all x, if x_7 cooks, then x_7 is listening to a song. There is at least one x for which x_7 is a police officer or x_7 will not listen to a song. For all x, if x_7 were a police officer, then x_7 is cooking. Therefore there is at least one x for which x_7 is a police officer if and only if x_7 will listen to a song.
forall x: P(x) -> Q(x). exists x: R(x) or ~Q(x). forall x: R(x) -> P(x). Therefore exists x: R(x) <-> Q(x).