Translate the following inference to logic notation: For all x, x_7 is not an office clerk or x_7 will listen to a song. There is at least one x for which x_7 is an office clerk or x_7 will write a letter. For all x, x_7 will play a game if and only if x_7 will write a letter. For all x, if x_7 listens to a song, then x_7 is playing a game. Therefore there is at least one x for which x_7 will play a game.
forall x: ~P(x) or Q(x). exists x: P(x) or R(x). forall x: S(x) <-> R(x). forall x: Q(x) -> S(x). Therefore exists x: S(x).