Let $\{X_0,...,X_n,...\}$ be a time-homogenous Markov chain, meaning $X_0,...,X_n,...$ are random variables (which we assume take values in a finite set $S$) such that we have
\begin{eqnarray*} P(X_n = y \ |X_0, X_1, . . . , X_{n−1}) &=& P(X_n = y \ |X_{n−1}) \text{ for all } n \ \ \ \ \text{(Markov property)} \\ P(X_{n+1}=x \ |X_n = y) &=& P(X_n = x \ |X_{n-1}=y) \text{ for all } n \ \ \ \ \text{(time homogeneity)} \\ \end{eqnarray*} In my professor's notes, I am reading that $$E(P(X_n = y|X_1) \ | X_0 = x) = E(P(X_{n-1} = y|X_1) \ |X_0 = x)$$ which supposedly follows from time homogeneity, but I don't follow. Namely, it seems that by "bumping down" the index from $n$ to $n-1$ for one variable, we should do the same and bump $X_1$ down to $X_0$. But somehow we're still conditioning on $X_1$ and not $X_0$. Help?