There are three events, $A$, $B$ and $D$. I know that $P(D)=0.2$, $P(A)=0.34$ and $P(B)=0.43$. I have calculated that $P(D\mid A)=0.5294$ and $P(D\mid B)=0.44186$. Now I need to calculate $P(D\mid (A\cap B))$, and I am stuck! I have tried expanding using Bayes' Theorem and I have had no success. One idea I had was that maybe $P(D\mid (A\cap B))=P(D\mid A)P(D\mid B)$ but I don't think this is generally true.
I also know that $P(A^c\mid D)=0.1$, $P(B^c\mid D)=0.05$, $P(A^c\mid D^c)=0.8$ and $P(B^c\mid D^c)=0.7$.
How can I go about calculating $P(D\mid (A\cap B))$ using what I have above?