It is the second law of Black Hole thermodynamics. Black Holes (BH) can only have their entropy stay the same or increase. A BH splitting into 2 would violate that.
BH entropy is proportional to the area of the horizon S ~ A, while the area is proportional to the horizon radius square, and the horizon radius (how far away from the center singularity) is proportional to its mass. So S ~ $M^2$. This is for a Schwarzschild BH. For a Kerr BH it's more complex involving the angular momentum, and if charged the charge also, but the results are similar.
As in the comment on Is there a way to split a black hole? in 2012, since $(m_1 + m_2)^2$ > $m_1^2 + m_2^2$, the end entropy is smaller than the starting entropy. Entropy decreases. That is impossible in BH thermodynamics. This is for Schwarzschild BH, a similar calculation is possible for Kerr and charged Kerr BHs (I've done the calculations for those to get the max grav radiation emitted, but not the other way around to prove they can't split, but I think it's tRue also). It is proven in one of the answers to the question in the 2012 reference above.
In all cases it also requires that energy is conserved, i.e., that there was not an external insertion of energy. If you do then it is possible
The impossible cases are the opposite of binary BH mergers which are possible and do happen, as detected by LIGO on 9/14/15. That increased entropy, splitting or the inverse would not be possible. In fact Hawking used the formulas for entropy for also the charged Kerr BHs, and derived the maximum gravitational radiation permitted in all cases, since any gravitational radiation carries energy, i.e.,part of the original mass M of the BH, and if it carries away too much the resulting horizons get too small and are prohibited by the second law. See the LIGO merger detection at http://www.ligo.org/science/Publication-GW150914/index.php