Prove that a composite number $n$ is a Carmichael number if and only if $b^{n-1} \equiv 1 \mod n$ for all integers b with $(b,n)=1$

Question

My book defines Carmichael numbers like this:

Let $n$ be a composite number. Then $n$ is said to be a Carmichael number if $b^n \equiv b \mod n$ for all integers $b$.

Question:

Prove that a composite number $n$ is a Carmichael number if and only if $b^{n-1} \equiv 1 \mod n$ for all integers b with $(b,n)=1$

I proved the only if part as follows: Let a composite number $n$ be a Carmichael number. Then for any integer $b$, $b^n \equiv b \mod n$. In particular, if $b$ is any integer with $(b,n)=1$, then $b^n \equiv b \mod n$. Thus $b^{n-1} \equiv 1 \mod n$. But I don't know how to prove the if part.