Thermodynamics proof for the distribution ratio exists for miscible solvents

It is well-known that the Nernst’s distribution law applicable for a solute distributes between two immiscible solvents. Thermodynamics proof for a solute distributes itself between two miscible solvents, until to get equilibria, the ratio of activity of a solute in two miscible solvents remains constant, but it’s possible only when the distribution ratio exists for miscible solvents.