I'm reading an unpublished paper in which the author makes the following conclusions several times:
Assumptions: finite probability space, $H$ is a random oracle, $X$ and $Y$ are two (not necessarily independent) random variables, $X \neq Y$ for every possible outcome.
Conclusion: $H(X)$ and $H(Y)$ are independent and uniformly distributed random variables.
"Proof" (not actually stated as such): obvious, $P(H(x) = a|D) = P(H(x) = a)$ for every event $D$.
Is this a valid conclusion? Are there any counterexamples which don't explicitly involve events $D$ similar to $H(x) \neq a$?