Translate the following inference to logic notation: Charles is happy if and only if Jessica is an astronaut. If Jessica were rich, then Charles is happy. For all x, x_19 is rich if and only if x_19 is an astronaut. Therefore Jessica is an astronaut if and only if Charles is happy.
p <-> P(a). Q(a) -> p. forall x: Q(x) <-> P(x). Therefore P(a) <-> p.