Expectation of the sum of two Dirac deltas

Question

Let $X_1,X_2$ be random vectors in $\mathbb{R}^n$ with joint PDF $f(x_1,x_2)$ such that $f(x_1,x_2)=f(x_2,x_1)\,\,\, \forall x_1,x_2 \in \mathbb{R}^{n}$ (i.e. $f$ is symmetric). Consider the following integral \begin{equation}\mathbb{E}[\delta_{X_1}(x)+\delta_{X_2}(x)]=\iint [\delta_{x_1}(x)+\delta_{x_2}(x)]\,f(x_1,x_2)\text{ d}x_1\text{d}x_2\end{equation} where $\delta_{x_i}(x)$ is the Dirac delta concentrated in $x_i$. I'm not sure if I have computed correctly such integration. My calculation are the following: \begin{equation}\begin{aligned} \mathbb{E}[\delta_{X_1}(x)+\delta_{X_2}(x)] &=\iint [\delta_{x_1}(x)+\delta_{x_2}(x)]\,f(x_1,x_2)\text{ d}x_1\text{d}x_2 \\ &=\iint \delta_{x_1}(x)\,f(x_1,x_2)\text{ d}x_1\text{d}x_2+\iint \delta_{x_2}(x)\,f(x_1,x_2)\text{ d}x_1\text{d}x_2 \\ &=\int f(x,x_2)\text{ d}x_2+\int f(x_1,x)\text{ d}x_1 \\ &=2\int f(x,w)\text{ d}w \end{aligned}\end{equation} where between the third and the fourth member I've exploited the sampling property of the deltas and between the fourth and the fifth member I've exploited the symmetry of $f$.