I would like to prove that $P(E,X|\hat{X})=P(E|\hat{X})P(X|E,\hat{X})$ where $X$,$\hat{X}$ and $E$ are three (not necessarily independent) RVs.
I tried to use Bayes' rule and the chain rule in the following manner:
$$P(E,X|\hat{X})=\frac{P(E,X,\hat{X})}{P(\hat{X})}=\frac{\require{cancel}\cancel{P(\hat{X})}P(X|\hat{X})P(E|X,\hat{X})}{\require{cancel}\cancel{P(\hat{X})}}=P(X|\hat{X})P(E|X,\hat{X})$$
But I am not sure how to proceed from here.