The following proofs follow the good proof technique summarized here.
Theorem: “the Weak Union” property of conditional independence: $(X \perp (Y,W) | Z) \Rightarrow (X \perp Y | Z,W)$ (see also equation 2.9 in Koller and Friedman, p. 25).
Proof
| Statement | Justification | |
| 1. | $X \perp (Y,W)|Z$ | Assumption |
| 2. | $P(X|(Y,W),Z) = P(X|Z)$ | Step 1, definition 2.3 & definition 2.4 (definitions of conditional independence for events and random variables respectively) |
| 3. | $P(X|Y,W,Z) = P(X|Z)$ | Ungrouping random variables |
| 4. | $X \perp Y | Z $ and $X \perp W|Z$ | Step 1, equation 2.8 (decomposition property of conditional independence) |
| 5. | $X \perp W | Z$ | Step 4, definition of conjuction |
| 6. | $P(X|W,Z) = P(X|Z)$ | Step 5, definition 2.3 |
| 7. | $P(X|Y,W,Z) = P(X|W,Z)$ | Step 2, Step 6, the transitive property of equality |
| 8. | $P(X|Y,(W,Z)) = P(X|(W,Z))$ | Step 7, Grouping random variables into sets of random variables |
| 9. | $X \perp Y | Z, W$ | Step 8, definition 2.3 & definition 2.4 |