Substitution of indifferent lotteries ($A\sim B⟹pA+(1-p)C\sim pB+(1-p)C$)

Apply the independence axiom for lottery preferences in both weak directions of the indifference relation. At $p=0$ the two mixtures are identical. Repeating this substitution and using transitivity lets one replace any finite collection of indifferent component lotteries, after which reduction of compound lotteries combines their outcome probabilities.