This was proved for SAT first: http://en.wikipedia.org/wiki/Cook%E2%80%93Levin_theorem
http://en.wikipedia.org/wiki/NP_complete
a. P is in NP, and
b. Every problem in NP can be converted to an instance of P.
Consider any two problems in NPC, P and Q. Then each is in NP, and each can be converted to the other. They are therefore equivalent.
QED.