# Proving conditional independence using a bayesian belief network / factorization

I have a bayesian belief network with 4 binary variables $$A, B, C, D$$. I now need to proof that for joint probability distributions factorized according the Bayesian network given below the conditional independency $$A⊥⊥D|C$$ always holds.

This by using factorization. Now I know that

$$p(A,B,C,D) = p(A)p(B)p(C|A,B)p(D|C)$$

but how can I go from that to proving that $$p(A,D|C) = p(A|C)p(D|C)$$ (definition conditional independence)

I would do it like this: \begin{align} P(A,D|C) &= P(A|D,C)P(D|C)\\ &=\frac{P(D,C|A)P(A)}{P(D,C)}P(D|C)\\ &=\frac{P(D|C,A)P(C|A)P(A)}{P(D|C)P(C)}P(D|C)\\ &=\frac{P(C|A)P(A)}{P(C)}P(D|C)\\ &=P(A|C)P(D|C) \end{align}
Where passages are just Bayes theorem and axioms of probability. I also used that $$P(D|C,A) = P(D|C)$$ because of redundancy of the conditioning.