Spreading out of an affine zero-set inclusion

Let $I,J\subseteq \mathbb{Z}[x_1,\dots ,x_n]$. If $V_{\mathbb{C}}(I)\subseteq V_{\mathbb{C}}(J)$, thenfor all but finitely many primes $p$. The Nullstellensatz puts a power of each generator of $J$ in $I\mathbb{Q}[x_1,\dots ,x_n]$; clearing denominators leaves only finitely many exceptional characteristics.