Looking for proof of conditional probability

From what I un­der­stand, the Kol­mogorov ax­ioms make no men­tion of con­di­tional prob­a­bil­ity. That is sim­ply defined. If I re­ally want to show how prob­a­bil­ity ac­tu­ally works, I’m not go­ing to ar­gue “by defi­ni­tion”. Does any­one know a mod­ified form that uses sim­pler ax­ioms than P(A|B) = P(A∩B)/​P(B)?