  We prove that if two semi-algebraic subsets of $\mathbb{Q}_p^n$ have the same
$p$-adic measure, then this equality can already be deduced using only some
basic integral transformation rules. On the one hand, this can be considered as
a positive answer to a $p$-adic analogue of a question asked by
Kontsevich-Zagier in the reals (though the question in the reals is much
harder). On the other hand, our result can also be considered as stating that
over $\mathbb{Q}_p$, universal motivic integration (in the sense of
Hrushovski-Kazhdan) is just $p$-adic integration.
