EGPWS has all the capabilities of GPWS and augments it by using a Global Positioning System, or GPS
Not necessarily. As Denti states, older types have a EGPWS box with an internal GPS. You have an interesting situation though types like the A320. These feed the EGPWS box with IRU data. If the airplane has GPS, then this works as you'd expect. However, if the airplane does not have GPS (old A320s), then EGPWS is dependent on proper FMS/IRS updating of position.
I remember an incident in Africa where a malfunctioning VOR almost caused a non-GPS A320 to fly into a hill, and the EGPWS function didn't say a thing.