Let's take for granted the following sampling property of Dirac's delta $\delta_w(x)\triangleq \delta(x-w)$ centered in $w$ \begin{equation*} \int_S \delta_w(x)\,f(w)\text{ d}w= \begin{cases} \quad \hfil f(x) & \text{ if } S\ni x \\ \quad \hfil 0 & \text{ otherwise} \end{cases} \end{equation*}
my question is the following: is it true the following integral simplification? \begin{equation*} \int_{S^2} \delta_{w_1}(x)\,f(w_1,w_2)\text{ d}w_1\text{d}w_2= \int_S f(x,w_2)\text{ d}w_2 \end{equation*}