A probability transformation

Question

Let $X,Y,Z$ be continuous random variables; $Z$ are independent of $X,Y$. Is the following transformation right ?
\begin{align} P(X,Y \in (a,b),Y+Z \notin (a,b))&=\int_a^bP(X \in (a,b),y+Z \notin (a,b))f_Y(y)dy\\ &=\int_a^bP(X \in (a,b))P(Z \notin (a-y,b-y))f_Y(y)dy. \end{align} Thank you !

Answer 1

The second line is right. The first line assumes independence of $X$ and $Y$ and is wrong otherwise. For instance, the left-hand side could be zero if $X$ and $Y$ never lie in $(a,b)$ at the same time, and yet the right-hand side could be non-zero.