This question is about a paragraph in The Analysis of Time Series: An Introduction with R (7th Edition) by Chatfield and Xing. I quote Section 12.1, p. 135:
In particular, suppose that $X_t$ follows an ARMA$(p, q)$ model described in Section 3.8, $$\phi(B)X_t = \theta(B)Y_t,$$ where $\phi(B)$ and $\theta(B)$ are polynomials of $B$ (NB: Backshift operator) with orders $p$ and $q$, respectively. Denote $\mathcal{F}_t$ as the set of observed data up to time $t$, i.e., $\{X_1, \ldots, X_t\}$. The observation $X_t$ can be expressed as $$X_t = \mu_t + Y_t \quad (12.1)$$ where $\mu_t$ is the mean of $X_t$ conditional on observed data $\mathcal{F}_{t-1}$ and given by $$\mu_t = E(X_t | \mathcal{F}_{t-1}) = \phi(B)X_t - (\theta(B) - 1)Y_t \quad (12.2)$$ and the innovation series $Y_t$ has mean 0 and conditional variance $$\sigma^2_t = \text{Var}(X_t | \mathcal{F}_{t-1}) = \text{Var}(Y_t | \mathcal{F}_{t-1}) \quad (12.3)$$ Volatility models discussed in the rest of this chapter deal with the evolution of $\sigma^2_t$ over time.
I am confused about Equation (12.2). How is this obtained? I can simplify the equation using \begin{align} \mu_t &= E(X_t | \mathcal{F}_{t-1}) \\ &= \phi(B)X_t - (\theta(B) - 1)Y_t \\ &= \phi(B)X_t - \theta(B)Y_t + Y_t \\ &= Y_t, \end{align}
where I use the defining equation $$\phi(B)X_t = \theta(B)Y_t,$$ but this does not make sense to me. Could someone please explain where (12.2) comes from? Thank you!