Let $W$ be a random variable that only takes on the values $1$ or $0$. Let $X$ and $Y$ be two other random variables. I came across the following: $$\mathbb{E}(Y|W=1, X)$$ How is this 'conditional expectation' even defined? It doesn't seem to fit the usual definition of a conditional expectation because $W=1$ is an event while $X$ is a random variable.
I was hoping to find a definition similar to that of $\mathbb{E}(Y|X)$, which has the property that $$\int_{B} \mathbb{E}(Y|X) dP=\int_{B} Y dP$$ for all $B \in\sigma(X)$.
Additional context:
Suppose we have observations $Y_i=W_i Y_i(1)+(1-W_i)Y_i(0)$ where the random variable $W_i \in\{0,1\}$ and the two random variables $Y_i(0), Y_i(1)$ can each be described by a regression model, that is $$Y_i(w)=\beta_{w,0}+\beta_{w,1}X_i+\epsilon_{w,i}, \hspace{5mm} w\in\{0,1\}$$ When the author writes $E(Y_i|W_i=1, X_i)$, I am pretty sure he means that $$E(Y_i|W_i=1, X_i)=\beta_{1,0}+\beta_{1,1}X_i$$ but I am wondering if this conditional expectation is well defined or made up notation.