Many books try to capture a proof as a sequence of inferences on statements. It looks natural at first because it is similar to how we make arguments in natural languages, but it is not natural as it looks.

If you try to formalize a proof in that way, the proof internally have to have a lot of backward references to indicate what inference uses what statements. This backward reference structure instantly becomes hugely convoluted, and it makes the proof cluttered and unreadable.