3-Digit Permutable Primes

Theorem
The $3$-digit permutable primes are:
 * $311, 199, 337$

and their anagrams, and no other.

Proof
It is confirmed that:
 * $113, 131, 311$ are all prime


 * $199, 919, 991$ are all prime


 * $337, 373, 733$ are all prime.

From Digits of Permutable Prime, all permutable primes contain digits in the set:
 * $\left\{ {1, 3, 7, 9}\right\}$

The sum of a $3$-digit repdigit number is divisible by $3$.

By Divisibility by 3 it follows that all $3$-digit repdigit numbers are divisible by $3$ and therefore composite.

Hence the possibly permutable primes are:
 * $113, 117, 119, 133, 137, 139, 177, 179, 199, 337, 339, 377, 379, 399, 779, 799$

and their anagrams.

We eliminate $113, 199, 337$ from this list, as it has been established that they are permutable primes.

Of those remaining, the following are composite:

It remains to demonstrate that at least one anagram of the remaining numbers:
 * $137, 139, 179, 379$

is composite.

We find that:

All contenders are eliminated except for the established permutable primes $113, 119, 337$ and their anagrams.