User blog comment:Pellucidar12/Peng-Pellucid-Armstrong Number/@comment-29915175-20170102011204/@comment-1605058-20170103093540

To emphasize the difference: if ZFC proves that "there exists a number \(n\) such that \(\varphi(n)\)", it is not necessarily true that ZFC proves \(\varphi(1)\) or \(\varphi(2)\) or \(\varphi(3)\) or ..., and similarly for "there exists a unique number...". A standard example for such a \(\varphi(n)\) is "\(n=1\) and CH holds or \(n=0\) and CH doesn't hold". It's clear precisely one of \(0,1\) satisfies \(varphi(n)\), but for neither of them ZFC can prove it, since CH is independent of ZFC.

For a more relevant example, ZFC easily proves that there is a unique number \(n\) such that \(n=\Sigma(10^10)\). However, for no specific integer \(n\) ZFC can prove that it is the value of \(\Sigma(10^10)\).