I think both theorems are indeed strongly related to each other, but we can’t just drop the second one. The second theorem seems more specific to me, as it specifies that the unique functional of the first theorem can be used to find the ground state energy related to the external potential. The second theorem is therefore more explicit. Due to the first theorem, we know that any other density should return a higher energy because of the one-to-one correspondence. So I wouldn’t say the second theorem is entirely independent from the first one, but I also wouldn’t say the second one is just a natural consequence of the first. Therefore, I only agree partially.