Just check everything carefully. If you've written your proof properly every step should follow logically from previous steps, and you should be able to check if yourself without issues. The problems happen when you're not critical enough of yourself, when the proof is very large and complex, or when you don't know your base axioms and the assumptions of your theorems well enough.

Proving the security of cryptographic protocols is a counterexample to this. Sometimes basic assumptions are very difficult to catch.