A non - rigorous proof:

The active forces and constarining forces acting on the particle are in equilibrium. Since the constraints are frictionless, the constraining forces are normal to the motion of the particle. Hence, the work done by the constraining force is zero. The total work done by all the forces is zero, since the resultant force is zero. Hence the work done by active forces on a virtual displacement consistent with the constraints is zero.

Now, assume that forces are not in equilibrium. Then by taking a virtual displacement in the direction of resultant force, some positive virtual work will be done. Hence, if the virtual work is zero, the particle will be in equilibrium.

 

 

Principle of Virtual work for rigid bodies

For a system of initially stationary rigid bodies, the necessary and sufficient condition for equilibrium is that the virtual work of the active forces be zero for all possible virtual displacements consistent with the constraint.