Translate the following inference to logic notation: If Elisabeth were an astronaut, then Elisabeth is taking a plane. For all x, if x_12 were an astronaut, then x_12 is taking a plane. If Elisabeth takes a plane, then Elisabeth is an astronaut. Therefore Elisabeth is taking a plane if and only if Elisabeth is an astronaut.
P(a) -> Q(a). forall x: P(x) -> Q(x). Q(a) -> P(a). Therefore Q(a) <-> P(a).