Suppose I have a fiber bundle $\pi: E\rightarrow B$, with fiber $F$, such that the Serre spectral sequence on cohomology is immediately degenerate. In other words, $H^*(E)=H^*(B)\otimes H^*(F)$.

I have a subvariety $X\subseteq E$ such that $\pi|_X$ is also a fiber bundle, with image $C$ and fiber $G$. I want to know the class (Poincare dual to the fundamental class) of $X$. Is it always true the $[X]=[C]\otimes [G]$? If so, could someone provide a reference?

(I would be happy with a reference in Chow rather than singular cohomology. If it matters, I'm actually interested in the case where $E$ is a flag variety $G/B$ and $B$ is $G/P$ for some parabolic, but $X$ is NOT a Schubert variety (where the desired fact is also a combinatorial fact about Schubert polynomials).)