In the Wikipedia article for the Rao-Blackwell theorem, the following step is used in the proof, namely a decomposition of the Rao-Blackwellized estimator's decomposition:
$$\mathbb{E}[(\delta_1(X) - \theta)^2] = \mathbb{E}[(\delta(X) - \theta)^2] - \mathbb{E}[\text{Var}(\delta(X) | T(X))]$$
I'm aware there's other questions regarding proofs of the Rao-Blackwell theorem, but none seem to be about this specific step. I've tried replacing $\delta_1$ with its expression in terms of $\delta$ and expanding the square to no avail...