Distributive generators of a lattice

Given a finite lattice $$L$$, suppose $$L$$ is generated by a set $$X$$ such that the distributive law holds for all $$a,b,c\in X$$ i.e. $$a\lor (b \wedge c) = (a\lor b)\wedge (a \lor c)$$.

Is $$L$$ distributive?