Translate the following inference to logic notation: For all x, x_13 is a carpenter if and only if x_13 is sad. There is at least one x for which if x_13 were a carpenter, then x_13 is playing a game. For all x, x_13 is sad or x_13 will not play a game. Therefore there is at least one x for which x_13 is a carpenter if and only if x_13 is sad.
forall x: P(x) <-> Q(x). exists x: P(x) -> R(x). forall x: Q(x) or ~R(x). Therefore exists x: P(x) <-> Q(x).