[Update: I see that the equality is wrong: Assume that p divide the right hand side. If p|(c,e) and p|(d,e), then p need not to divide the left hand side. Thank you all for your comments]
Let $a,b,c,d,e$ be integers. I am digging the following equality:
$\Big[ (a,b) , (a,c,e), (b,d,e) \Big] = \Big( \big[a, (d,e) \big] , \big[b,(c,e) \big] \Big) $
where $[\ldots]$ and $(\ldots)$ denote LCM and GCD respectively. I can show that this equality holds as follows: take a prime $p$ that divide the left side, then it will also divide the right side, and vice versa. But is there another elegant way to show that this equality holds, for instance, by using the properties of GCD and LCM?