Showing $ \int_a^bf(x)dx=-\int_{b}^{a} f(x)dx$, without the Fundamental Theorem of Calculus

Question

Without using the Fundamental Theorem of Calculus, I want to show that $$ \int_a^bf(x)dx=-\int_{b}^{a} f(x)dx $$ when $b < a$. Apparenly it follows from the rule $$ \int_a^c f(x)dx=\int_a^b f(x)dx+\int_b^c f(x)dx $$ which holds for all real numbers $a, b \ \mathrm{and} \ c$ by setting $c=a$. But where does this rule come from? Or is it some kind of axiom of integrals? And don't we also have to define separately that $$ \int_a^a f(x)dx=0? $$