 Birkhoff’s theorem is the converse: for any class \(C\) such that \(P(C) = S(C) = H(C), C\) is a variety. In fact the theorem is slightly stronger: for any class \(C\), HSP\((C)\) is a variety. That is, to construct all the models of the theory of \(C\) it suffices to close \(C\) first under direct products, then under subalgebras, and finally under homomorphic images; that is, later closures do not compromise earlier ones provided \(P, S\), and \(H\) are performed in that order.