How do we know that LCM works the same as Phi for the RSA cryptosystem?

Question

In the case of Euler's phi function, we know from the Euler generalization of Fermat's little theorem that $M^{\phi(n)} \equiv 1$ mod n. However a lot of modern RSA implementations use LCM ($\lambda(p-1,q-1)$), where $pq = n$, instead of $\phi(n)$. How do we know that any $d$ that is coprime with LCM(p-1,q-1) will produce $m^{ed} \equiv m$ mod n?