Am I a Pillai prime?
Python 2, 115 111 110 109 bytes

-6 bytes thanks to Mr. Xcoder


lambda n:n&gt;2and cmp(*map(all,zip(*[[n%x==1or~f(x)%n,n%x]for x in range(2,n)])))&lt;0
f=lambda x:0**x or x*f(x-1)


Try it online!

The functions consist of two parts ~-n%x&lt;1or~f(x)%n&gt;0 that verifies if n don't satisfies the "Pillai conditions", and n%x&gt;0 for the prime validation.
After that all is applied to both items, the first item will contain False/0 if there is a valid "Pillai number", and the second will contain True/1 if n is prime.
These are passed to cmp that will return -1 in this cenario (is a valid Pillai prime). The other combinations [[0, 0], [1, 0], [1, 1]] will return 0 or 1