Translate the following inference to logic notation: Elisabeth is an astronaut if and only if Elisabeth is climbing a mountain. For all x, z_2 will climb a mountain if and only if z_2 is an astronaut. Therefore Elisabeth is climbing a mountain if and only if Elisabeth is an astronaut.
P(a) <-> Q(a). forall x: Q(x) <-> P(x). Therefore Q(a) <-> P(a).