Proof of Dixon's identity only using Chu-Vandermonde

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.

• I don't know of such a proof in the literature. This question has now been open for some time, and I would be interested to see your proof using only Chu–Vandermonde. – Mark Wildon Aug 27 '18 at 16:08
• It's part of a proof or more general identities which I am including in a paper in preparation. I will definitely let you know when the paper is available. – Abdelmalek Abdesselam Aug 29 '18 at 21:19
• Is the paper in preparation now available? – LSpice Feb 4 at 12:38
• @MarkWildon: Sorry for the wait and thank you for your patience. The paper containing the said proof is now posted at arxiv.org/abs/1903.11147 – Abdelmalek Abdesselam Mar 28 at 13:51
• @LSpice: Sorry for the wait and thank you for your patience. The paper containing the said proof is now posted. See my previous comment to Mark for the link. – Abdelmalek Abdesselam Mar 28 at 13:52