Tough set of 8's

If:

${ 8 }^{ A }+{ 8 }^{ B }+{ 8 }^{ C }+{ 8 }^{ C }\quad =\quad \overline { ABCC }$

A + B + C + C = P

Where ABCC represents the four digit representation of a prime number, and P is the sum of A,B,C, and C, which also happens to be prime.

Then what is A + B + C ?

Details and assumptions:

ABCC represents a four digit number. In no way does ABCC represent multiplication. ABCC and P are both prime numbers. ABCC is positive.