# Wick's Theorem for Yukawa Theory

I'm studying Quantum Field Theory and encountered Wick's theorem: for the real Klein Gordon field $\phi(x)$, one has

$$T(\phi(x_1)\cdots\phi(x_n)) = N(\phi(x_1)\cdots\phi(x_n) + \text{ all possible contractions})$$

where $T$ denotes time ordering and $N$ denotes normal ordering. With suitable definitions of $T$ and $N$ this can be extended to Dirac fields $\psi(x)$ as well:

$$T(\psi_{a_1}(x_1)\cdots\psi_{a_n}(x_n)) = N(\psi_{a_1}(x_1)\cdots\psi_{a_n}(x_n) + \text{ all possible contractions})$$

where the $a_i$ are spinor indices. But what a lot of books/notes on the topic do is the following. They prove the theorem for the KG field, and later for the Dirac field, and then they suddenly apply it to Yukawa theorey, e.g., to mixed expressions of the form

$$T(\psi_a(x_1)\bar{\psi}_b(x_2)\phi(x_3)\bar{\psi}(z)\psi(z)\phi(z)) = \dots$$

I guess the theorem must also hold for these kind of expressions, but I have not been able to find a reference that proves it. I understand the (canonical) proofs for the KG and Dirac case seperately, but I don't see how one would extend these to mixed expressions. If anyone knows how to prove the theorem for these cases, or knows a reference to such a proof, I would love to hear (well, read) it.