Let $(\Omega, \mathcal{F}, P)$ be a probability space. Suppose $X$ is a random variable with distribution function $F_{X}$, and $A$ an event on $(\Omega, \mathcal{F}, P)$. Then the law of total probability states $$P(A)=\int_{-\infty}^{\infty} P(A \mid X=x) d F_{X}(x) .$$ If $X$ admits a density function $f_{X}$, then the result is $$P(A)=\int_{-\infty}^{\infty} P(A \mid X=x) f_{X}(x) d x. $$ Moreover, for the specific case where $A=\{Y < y \}$, where $y \in \mathbb R$, then this yields $$P(Y < y )=\int_{-\infty}^{\infty} P(Y <y \mid X=x) f_{X}(x) d x .$$
Is there a total probability law for quantile functions instead of cumulative probability distributions?
For instance, if $Y|X$ is distributed according to $\mathcal N (X,1)$, can we write the quantile function of $Y$ as a functional of the quantile function of $\mathcal N (X,1)$ ?