Translate the following inference to logic notation: For all x, if x_1 were a carpenter, then x_1 is fast. There is at least one x for which if x_1 were fast, then x_1 is climbing a mountain. For all x, if x_1 climbs a mountain, then x_1 is a carpenter. Therefore there is at least one x for which x_1 is fast if and only if x_1 is a carpenter.
forall x: P(x) -> Q(x). exists x: Q(x) -> R(x). forall x: R(x) -> P(x). Therefore exists x: Q(x) <-> P(x).