For any integers $a,b,c\ge 0$, one has the well known identity or Dixon's Theorem: $$ \sum_{k\in\mathbb{Z}} (-1)^k \left(\begin{array}{c}a+b\\a+k\end{array}\right) \left(\begin{array}{c}b+c\\b+k\end{array}\right) \left(\begin{array}{c}c+a\\c+k\end{array}\right) \ =\ \frac{(a+b+c)!}{a!\ b!\ c!}\ . $$ There are tons of proofs in the literature, but I am wondering if there is one which only uses (possibly repeated) applications of the Chu-Vandermonde identity $$ \sum_{j\ge 0} \left(\begin{array}{c}a\\j\end{array}\right) \left(\begin{array}{c}b\\c-j\end{array}\right) \ =\ \left(\begin{array}{c}a+b\\c\end{array}\right)\ . $$
I should add, for more context, that I found such a proof and was wondering if something of that kind appeared in the literature. I would be very surprised if it didn't. A good example of what I mean by a proof only using Chu-Vandermonde is the derivation of the single index formula for 6j symbols by Racah in appendix B of his 1942 article "Theory of complex spectra II". BTW, appendix A of the same article also contains a proof of the above identity.
Edit (Mar 28, 2019):
The proof that I found is contained in Section 3 of the paper: "An algebraic independence result related to a conjecture of Dixmier on binary form invariants" which is now available. Given that I did not get an answer to this question since it was posted a few months ago, I would tend to think that my proof is new. Again, if you know otherwise, please point me towards the relevant reference.
