By definition, $\mathbb{P}(E \mid \mathcal{G}) : = \mathbb{E}(I_E \mid \mathcal{G})$. Applying results

(i)	R2150: Expectation of conditional expectation for a random euclidean real number
(ii)	R2089: Unsigned basic expectation is compatible with probability measure

one then has \begin{equation} \mathbb{E}(\mathbb{P}(E \mid \mathcal{G})) = \mathbb{E}(\mathbb{E}(I_E \mid \mathcal{G})) = \mathbb{E}(I_E) = \mathbb{P}(E) \end{equation} $\square$