I just learned about conditional expectation this week and I am a bit confused on understanding $E(E(X|\mathcal{F}))=E(X)$ where $(\Omega,\mathcal{F}_0,P)$ is a probability space, $X:\Omega\to\mathbb{R}$ is a random variable and $\mathcal{F}\subset\mathcal{F}_0$ is a sub $\sigma$-algebra. The proof is straightforward as the following: by definition of conditional expectation we have for all $A\in\mathcal{F}$, $\int_A X dP=\int_A E(X|\mathcal{F}) dP$. Take $A=\Omega$, then we have the result as desired.
However, I am stuck intuitively that $E(X|\mathcal{F})$ represents a random variable or a "best guess" given information/observables in terms of $\mathcal{F}$. Why did those information disappear when we try to find the best guess of this best guess or the mean of this random variable.
I have one discrete example in mind: consider rolling a dice where $\Omega=\{1,2,\ldots,6\}$ and $X$ reads the outcome. Let $\mathcal{F}_0$ be the set of all subsets and $\mathcal{F}=\sigma(\{1,3,5\},\{2,4,6\})$ i.e. $\mathcal{F}$ contains or can observe parity information. Now, $E(X|\mathcal{F})= 3$ if $X$ odd and $4$ if $X$ even and thus $E(E(X|\mathcal{F}))=3.5$ which equals $E(X)$. However, I still don't understand why the given information disappeared.
Thanks!