tower property with respect to filtration and measurable event

Question

Let $E$ an event measurable with respect to a $\sigma$-algebra $\mathcal{F}$, i.e. such that $\mathcal{G}=\sigma(E)\subset\mathcal{F}$. Let $X$ a random variable. I wanted to see if $\mathbb{E}[\mathbb{E}_{\mathcal{F}}X|E]=\mathbb{E}(X|E)$, thus generalising the usual property ($\mathbb{E}[\mathbb{E}_{\mathcal{F}}X]=\mathbb{E}X$) to expectations conditional on more specific measurable events than the trivial ones. I thought of applying the tower property with repsect to $\mathcal{G}$: we know that $\mathbb{E}(X|\mathcal{F}|\mathcal{G})=\mathbb{E}_{\mathcal{G}}(X)=\mathbb{1}_{E}\mathbb{E}(X|E)+\mathbb{1}_{E^c}\mathbb{E}(X|E^c)$ and similarly $\mathbb{E}(X|\mathcal{F}|\mathcal{G})=\mathbb{E}_{\mathcal{G}}(\mathbb{E}_\mathcal{F}X)=\mathbb{1}_{E}\mathbb{E}(\mathbb{E}_\mathcal{F}X|E)+\mathbb{1}_{E^c}\mathbb{E}(\mathbb{E}_\mathcal{F}X|E^c)$. So wouldn't this, by taking $\omega\in E$, yield $\mathbb{E}(X|E)=\mathbb{E}(\mathbb{E}_\mathcal{F}X|E)$? I just wanted to check if my attempt is correct. Thanks!