In class we went over a proof to show that $$(X_{1}, X_{2}) \perp X_{3} \equiv X_{1} \perp X_{3} | X_{2}$$ for 3-way contingency tables. I understand the rest of the proof, but can't see how these were derived: $$LHS \Longrightarrow P(X_{1} = i, X_{2} = j, X_{3} = k) = P(X_{1} = i, X_{2} = j)P(X_{3} = k)$$ and $$RHS \Longrightarrow \frac{P(X_{1} = i, X_{2} = j, X_{3} = k)}{P(X_{2} = j)} = \frac{P(X_{1} = i, X_{2} = j)}{P(X_{2} = j)} \frac{P(X_{2} = j, X_{3} = k)}{P(X_{2} = j)}$$
I assume Bayes formula has been used, but am unsure where or how. Any help would be awesome!