Translate the following inference to logic notation: There is at least one x for which if x_12 works, then x_12 is rich. For all x, x_12 will write a letter if and only if x_12 will work. For all x, x_12 is rich if and only if x_12 will write a letter. Therefore there is at least one x for which x_12 will write a letter if and only if x_12 is rich.
exists x: P(x) -> Q(x). forall x: R(x) <-> P(x). forall x: Q(x) <-> R(x). Therefore exists x: R(x) <-> Q(x).