If a lattice $L$ is distributive then it can be shown that for $a,b,c\in L$: $$[a\wedge b=a\wedge c\text{ and }a\vee b=a\vee c]\implies b=c$$
So for fixed $a,u,v\in L$ there is at most one $b$ such that $u=a\wedge b$ and $v=a\vee b$.
Is the converse of this true? More precisely, is the following statement true?
If for each triple $a,u,v$ in a lattice there is at most one $b\in L$ such that $u=a\wedge b$ and $v=a\vee b$ then the lattice is distributive.