Perfect normality implies hereditary normality

For disjoint closed sets $A,B$ in a subspace $Y$, their ambient closures $C,D$ satisfy $(C\cap D)\cap Y=\varnothing$. In a perfectly normal space choose continuous functions $f,g$ with exact zero sets $C,D$. The function $f/(f+g)$ is continuous on $X\setminus (C\cap D)$, which contains $Y$, and separates $A,B$. Disjoint inverse images of endpoint neighborhoods give normality of $Y$.