Translate the following inference to logic notation: For all x, x_7 is excited or x_7 is a cashier. For all x, if x_7 plays a game, then x_7 is cooking. For all x, x_7 is not excited or x_7 will play a game. There is at least one x for which x_7 will cook if and only if x_7 is a cashier. Therefore there is at least one x for which x_7 will cook.
forall x: P(x) or Q(x). forall x: R(x) -> S(x). forall x: ~P(x) or R(x). exists x: S(x) <-> Q(x). Therefore exists x: S(x).