Is it possible to derive the invariant spacetime interval from Einstein's two postulates for SR?

Question

In many textbooks, the interval

$$ I = -(c\Delta t)^2 + (\Delta x)^2 + (\Delta y)^2 + (\Delta z)^2 $$

is taken for granted as the same for two events in any reference frame.

Is it possible to derive this just from the two postulates,

1. That the laws of physics are the same in all reference frames
2. That the speed of light c is constant for all reference frames?