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

(i)	R2158: Conditional expectation given independent sigma-algebra
(ii)	R2089: Unsigned basic expectation is compatible with probability measure

we have \begin{equation} \mathbb{P}(E \mid \mathcal{G}) = \mathbb{E}(I_E \mid \mathcal{G}) \overset{a.s.}{=} \mathbb{E}(I_E) = \mathbb{P}(E) \end{equation} $\square$