See Proposition 11.5 (and the discussion leading up to it) in Kanamori's book The Higher Infinite. Note that Kanamori states the result a bit more optimally: if $M\models\mathsf{ZF}$ + "$\omega_1$ is regular" + $\mathsf{PCP}$ then $\omega_1^M$ is inaccessible in $L^M$ (and in fact $M\models$ "$\omega_1$ is inaccessible to reals," that is, $\omega_1^M$ is inaccessible in $L[a]^M$ for every $a\in\mathbb{R}^M$).
Since it's pretty short, I'll sketch the proof here for completeness:
Suppose $M$ satisfies the necessary conditions; from now on, all statements are made within $M$.
First, we prove an auxiliary result: that $\omega_1^{L[a]}$ being countable for each real $a$ implies that $\omega_1$ is inaccessible to reals. We prove the contrapositive. Suppose $\omega_1$ is not inaccessible in $L[a]$ for some real $a$. By the usual condensation argument (relativized) we have $L[a]\models\mathsf{GCH}$, so we must have $\omega_1=\rho^{+^{L[a]}}$ for some $L[a]$-cardinal $\rho$. Now $\rho$ is countable in reality, hence coded by a real; let $b$ be a real coding a well-ordering of $\omega$ isomorphic to $\rho$. Then $a\oplus b$ is a real with $\omega_1^{L[a\oplus b]}\ge\omega_1$.
Now suppose $\omega_1$ is not inaccessible to reals. By the above, this means that $\omega_1^{L[a]}=\omega_1$ for some real $a$. But per Godel we know that $L[a]\models\omega_1\le 2^{\aleph_0}$, or put another way there is an injection from $\omega_1^{L[a]}$ into the reals. Sicne $\omega_1^{L[a]}=\omega_1$, this means there is an injection from $\omega_1$ into the reals, whence (by Bernstein) $\mathsf{PSP}$ fails.
