Quasi-isomorphism at every filtration level implies isomorphic spectral sequences?

Question

Let $f:C\to C^\prime$ be a chain map. Suppose there are increasing filtrations $F_p$, $F^\prime_p$ ($p\ge0$) on $C$, $C^\prime$ respectively, such that for each $p$, $f(F_pC)\subset F^\prime_pC^\prime$ and $f:F_pC\to F^\prime_pC^\prime$ is a quasi-isomorphism. Then does $f$ induce an isomorphism $E^r_{p,q}(C)\cong E^r_{p,q}(C^\prime)$ for $r$ large (say $r=1$ or $r=2$)?