I highly recommend the courses on brilliant.org. They are not axiomatic, but the...

I highly recommend the courses on brilliant.org. They are not axiomatic, but they heavily focus on building up deep insight rather than mechanical problem solving -- https://brilliant.org.

That said, for a rigorous proof-based approach to high school math, you may enjoy "Basic Mathematics" by Lang: https://www.amazon.com/Basic-Mathematics-Serge-Lang/dp/03879...