Modular arithmetic

[Stereotomy]

\( ^{n}a\text{ mod }b= {}^{m}a\text{ mod }b \)

\( \text{For }^{n}a,{}^{m}a > b,\text{ }a,b,n,m \in \mathbb{N} \)

Is this actually true? And if so is there a proof of it I'll be able to wrap my mind around?

I dont think it is true. For example:
\( 7^7 \text{mod} 5 = 3\neq 2 = 7 \text{mod} 5 \)

Ah, good point, though
\( 7^{7^{7}} \text{mod} 5 = 3 \)
Is true as well. In fact, thinking about it, the numbers I tried out with this all had b>a. Perhaps that's an additional condition that either b > a or m, n > 1?

Just quickly tried this for a few low examples, a = 8, 9, 10, 11, and it seems to hold.