== "Helfgott's claim covers both versions of the conjecture." == 

 Perhaps it is better to say that he proved the version about odd primes, as the version with any primes follows directly from that (just note as the last sentence of your proof that 7 = 2 + 2 + 3). If he had proven the version about any primes, however, the version about odd primes would not follow from that (as then perhaps we could express a number as the sum of three primes in only one way, but two of these primes were 2).