Bayes Theorem: expressing joint probability in terms of chained conditional probability

Question

I would like to check whether my understanding of the Bayes theorem is correct using the following exercise.

Suppose we have 4 random variables A,B,C,D. We are given the joint probabilities for each pair of variables i.e. p(A,B), p(B,C) etc. and the conditional probability p(A,B|C,D). We are supposed to calculate p(A,C|B,D).

1. The Bayes formula: p(B|A) = p(A,B) / p(A) = p(A|B)p(B) / p(A)
2. The solution for the above problem: p(A,C|B,D) = p(A,B,C,D) / p(B,D)
3. Developing 2): p(A,C|B,D) = p(A,B|C,D)p(C,D) / p(B,D) which is the official solution

My question: Can I express the joint probability p(A,B,C,D) for example by p(A,D|B,C)p(B,C) or p(C,D|A,B)p(A,B) if we were given a different conditional probability? Thank you for any answers :)