The fringe width (β) is defined as the distance between two consecutive bright fringes:
$\mathit{\beta }={\mathit{y}}_{\mathit{n}+1}-{\mathit{y}}_{\mathit{n}}=\frac{\left(\mathit{n}+1\right)\mathit{\lambda }\mathit{D}}{\mathit{d}}-\frac{\mathit{n}\mathit{\lambda }\mathit{D}}{\mathit{d}}=\frac{\mathit{\lambda }\mathit{D}}{\mathit{d}}$
Similarly, the distance between two consecutive dark fringes:
$\mathit{\beta }=\mathit{y}{\prime }_{\mathit{n}+1}-\mathit{y}{\prime }_{\mathit{n}}=\frac{\left(2\mathit{n}+1\right)\mathit{\lambda }\mathit{D}}{2\mathit{d}}-\frac{\left(2\mathit{n}-1\right)\mathit{\lambda }\mathit{D}}{2\mathit{d}}=\frac{\mathit{\lambda }\mathit{D}}{\mathit{d}}$
Since both expressions give the same value, the fringe widths for bright and dark fringes are equal.