Mathematically is just the chain rule? Since

$\dfrac{\partial}{\partial \vec{r}_i}V_{ij}(\vert \vec{r}_i-\vec{r}_j\vert)=\dfrac{\partial V_{ij}}{\partial (\vec{r}_i-\vec{r}_j)}\dfrac{\partial (\vec{r}_i-\vec{r}_j)}{\partial \vec{r}_i}=\dfrac{\partial V_{ij}}{\partial (\vec{r}_i-\vec{r}_j)}(1-\delta_{ji})$

$\dfrac{\partial}{\partial \vec{r}_j}V_{ij}(\vert \vec{r}_i-\vec{r}_j\vert)=\dfrac{\partial V_{ij}}{\partial (\vec{r}_i-\vec{r}_j)}\dfrac{\partial (\vec{r}_i-\vec{r}_j)}{\partial \vec{r}_j}=\dfrac{\partial V_{ij}}{\partial (\vec{r}_i-\vec{r}_j)}(\delta_{ij}-1)=-\dfrac{\partial}{\partial \vec{r}_i}V_{ij}(\vert \vec{r}_i-\vec{r}_j\vert)$

As far as $\delta_{ij}=\delta_{ji}.$

Mathematically is just the chain rule? Since

\begin{align} \dfrac{\partial}{\partial \vec{r}_i}V_{ij}(\vert \vec{r}_i-\vec{r}_j\vert)&=\dfrac{\partial V_{ij}}{\partial (\vec{r}_i-\vec{r}_j)}\dfrac{\partial (\vec{r}_i-\vec{r}_j)}{\partial \vec{r}_i}\\ &=\dfrac{\partial V_{ij}}{\partial (\vec{r}_i-\vec{r}_j)}(1-\delta_{ji}) \end{align}

$$$$

\begin{align} \dfrac{\partial}{\partial \vec{r}_j}V_{ij}(\vert \vec{r}_i-\vec{r}_j\vert)&=\dfrac{\partial V_{ij}}{\partial (\vec{r}_i-\vec{r}_j)}\dfrac{\partial (\vec{r}_i-\vec{r}_j)}{\partial \vec{r}_j}\\&=\dfrac{\partial V_{ij}}{\partial (\vec{r}_i-\vec{r}_j)}(\delta_{ij}-1)\\&=-\dfrac{\partial}{\partial \vec{r}_i}V_{ij}(\vert \vec{r}_i-\vec{r}_j\vert) \end{align}

As far as $\delta_{ij}=\delta_{ji}.$