Translate the following inference to logic notation: For all x, if x_3 cooks, then x_3 is writing a letter. For all x, if x_3 were happy, then x_3 is cooking. There is at least one x for which x_3 is happy if and only if x_3 will drink water. Therefore there is at least one x for which if x_3 drinks water, then x_3 is writing a letter.
forall x: P(x) -> Q(x). forall x: R(x) -> P(x). exists x: R(x) <-> S(x). Therefore exists x: S(x) -> Q(x).