I went over the proof and discovered something else that I typed incorrectly. It must be that posting proofs online makes me nervous.
Here is the uniqueness proof
again, reworded slightly:
c is smallest, so none of the unknowns except c can possible be the smallest prime; therefore, a,b,d and e must all be odd, because only the smallest prime is even.
c must be even, because it is the difference between two odd numbers.
c must equal 2, because it is both even and prime.
The simpler equations tell us that d + 2 = b, and b + 2 = a; so d, b and a must be three consecutive odd numbers.
The only time you find three consecutive odd numbers that are all prime is with 3, 5 and 7. Above that level when you have three consecutive odds, one of them will always be composite (i.e. not prime), because it will be divisible by 3. So we know that a=7, b=5, and d=3.
This just leaves us with 7 * 5 - 2 = 3 * e. The only solution to that is e = 11.
I have just shown that if your conditions hold then the answer is "a=7, b=5, c=2, d=3, e=11." The contrapositive of that statement is, if the answer is not "a=7, b=5, c=2, d=3, e=11", then your conditions do not all hold.
In other words, "a=7, b=5, c=2, d=3, e=11" is the only acceptable answer. It is therefore the unique solution.