[MrEldritch]

Incidentally, fun fact - you already know that X, Y, and Z linear momentum are all conserved separately. In special relativity, mass-energy conservation gets folded into this as well - an object's mass-energy is just the component of linear momentum along the time axis! (and "rest mass" is its value in the reference frame where the object is otherwise stationary and moving only through time.)

[antidesitter]

This is conservation of 4-momentum, whose spatial components are ordinary momentum and whose temporal component is energy. By Noether’s theorem, conservation of 4-momentum is due to the invariance of physical laws under 4-translations (spatial and temporal).