How is it that the equilibrium constant does not depend on the mechanism?

For a reaction of the form

$$\ce{aA + bB <=> cC + dD}$$

the equilibrium constant is

$$K_c=\frac{[\ce{C}]^c[\ce{D}]^d}{[\ce{A}]^a[\ce{B}]^b}$$

regardless of the mechanism of the reaction. Why is this the case?

I've seen derivations that use a reaction in one elementary step to demonstrate this, but this obviously doesn't work in general. The sample reaction used was $\ce{N2O4 <=> 2NO2}$.