While computing the price at time $t$ of a European call option with strike $K>0$ and maturity $T>0$ for $t$ in $[0,T]$, I encountered the following conditional expectation which I cannot compute ($\sigma, r>0$ are just constants):
$$\mathbb{E}\left[\left.\max\left\{e^{-\frac{1}{2}\sigma^2(T-t)+\sigma(W_T-W_t)}-\frac{Ke^{-rT}}{e^{-\frac{1}{2}\sigma^2 t+\sigma W_t}},0\right\}\right|\mathfrak{F}_t\right]$$
Here $\{W_t\}_{t\ge 0}$ is a Brownian motion which is adapted to the filtration $\mathfrak{F}$. The lecture notes I am studying suggest now to set $$Y:=\frac{Ke^{-rT}}{e^{-\frac{1}{2}\sigma^2 t+\sigma W_t}},$$ which is an $\mathfrak{F}_t$-measurable random variable and to compute the expectation $$\mathbb{E}\left[\max\left\{e^{-\frac{1}{2}\sigma^2(T-t)+\sigma(W_T-W_t)}-Y,0\right\}\right]$$ while treating $Y$ as a constant. The latter problem can be easily solved (albeit some lengthy computations occur). The notes say that this approach is valid since the random variable $W_T-W_t$ is independent of $\mathfrak{F}_t$. The latter fact is clear to me. The entire computation would also be clear to me if $Y$ really were a constant which is definitely not the case.
So, having in mind the properties of conditional expectations (see e.g. here), I am wondering why this approach is valid. Does someone have an idea?