Translate the following inference to logic notation: For all x, y is a carpenter or y will sleep. For all x, y is a carpenter if and only if y is a lawyer. For all x, if y were a lawyer, then y is happy. There is at least one x for which if y sleeps, then y is happy. Therefore there is at least one x for which y is happy.
forall x: P(x) or Q(x). forall x: P(x) <-> R(x). forall x: R(x) -> S(x). exists x: Q(x) -> S(x). Therefore exists x: S(x).