Modern logic and set theory meets some of the requirements of this. At the time, he did not have for-all, exists, and so on. Leibnitz himself created some proof

Modern logic and set theory meets some of the requirements of this. At the time, he did not have for-all, exists, and so on. Leibnitz himself created some proofs with the limited logic of his day. Nowadays much more can be done. It's not as all-encompassing, but it is a step forward.