Note that there’s nothing logically wrong with having axioms that follow from other axioms. They’re just dependent, and you’ve written more than you need to. The theory is not invalidated in the least.
For the curious, I explained G’s theorem in great detail in a recent GD thread. Search on “inconsistent incomplete” within the past three months. I’ll try to dig up a link, but the search engine’s being slow. Once I find the thread, the discussion is in the last few posts, on page 2. Note that it presupposes that you’re familiar with predicate logic–this may be a small barrier for some.