Suppose and arc the slits at a distance from each other.
Distance of screen be . Let be a point where there is a bright fringe.
A glass plate is placed in the path of the ray from to .
We know that the path difference between the rays in absence of glass plate is
In presence of the glass plate, the optical path length of the ray from to will be different.
The total optical path length for this ray is given by

Where is the refractive index of the glass platc and is its thickness.
Hence the new path difference is given by

For a bright fringe,
and distance of the bright fringe from the central fringe

Hence the fringe width remains constant.