Translate the following inference to logic notation: For all x, if x_4 were a janitor, then x_4 is listening to a song. For all x, if x_4 plays a game, then x_4 is an astronaut. For all x, x_4 will write a letter or x_4 will play a game. For all x, if x_4 listens to a song, then x_4 is an astronaut. There is at least one x for which x_4 will not write a letter or x_4 is a janitor. Therefore there is at least one x for which x_4 is an astronaut.
forall x: P(x) -> Q(x). forall x: R(x) -> S(x). forall x: T(x) or R(x). forall x: Q(x) -> S(x). exists x: ~T(x) or P(x). Therefore exists x: S(x).