Given that $X_0=3$ and a discrete time Markov Chain with transition probabilities $p(i,j)$, I need to express $$P(X_6=7, X_5=3 \mid X_4=1, X_9=3)$$ in terms of the transition probabilities $p(i,j)$.
So far I have that \begin{align} P(X_6=7, X_5=3 \mid X_4=1, X_9=3) &= P(X_6=7, X_5=3 \mid X_4=1)\\ &=P(X_6=7 \mid X_5 = 3, X_4=1)\, P(X_5=3 \mid X_4=1)\\ &= P(X_6=7 \mid X_5=3) \, P(X_5=3 \mid X_4=1). \end{align} However I'm not sure where to go from here. Specifically, I'm not sure how to use the fact that $X_0=3$.