Let $X_1$, $X_2$ and $X_3$ be three i.i.d. Bernouilli random variable of parameter $p$. Let $X=(X_1,X_2,X_3)$ and $X'=(X_1,X_3,X_2)$ and let $f:\{0,1\}^3\rightarrow \mathbb{R}$ be a function.
I would like to prove that: $$\mathbb{E}[\mathbb{E}[f(X)|X_1+X_2] (f(X')-f(X))]\leqslant 0$$
Without the conditional expectation, it is easy to prove the same statement since the $X_i$'s are exchangeable (and therefore $X \sim X'$), i.e.,
\begin{align} \mathbb{E}[f(X)(f(X')-f(X))] &=\mathbb{E}[f(X')(f(X)-f(X'))] \\ &=-\frac{1}{2} \mathbb{E}[(f(X)-f(X'))^2] \end{align} It feels like the result should be true (or at least, I would like it to be true), but I am having a hard time using the conditional expectation.