The Riemann-Roch without denominators can be expressed as follows: Let $f: X\rightarrow Y$ be a closed embedding of quasi-projective smooth $k$-varieties of codimension $d$ for some field $k$. Let $E$ be a vector bundle of rank $r$ on $X$. Then we have in $\mathrm{CH}^{\bullet}(Y)$ that \begin{equation*} c_{j}(f_{*}([E])) = \begin{cases} 0, & \text{if $0<j<d$}; \\ r(-1)^{d-1}(d-1)!\cdot[X], & \text{if $j=d$}. \end{cases} \end{equation*} This can be found in Fulton's book 'Intersection Theory', Theorem 15.3 and the examples following it. My question is:

Does this hold when $X$ is just a closed (not smooth) subvariety of $Y$ ($Y$ is still smooth)? I mainly want to know is it true that $$c_{j}(f_{*}([E])) = 0 \ \text{if $0<j<d$}?$$

I want to use this to show that the $p$-th Chern class maps the $p+1$-th filtration of $K$-group to $0$, as claimed in Example 15.3.6 in that book. All the sources I found don't mention this point or rather sloppy about this. Could you explain this, or give a proper reference? Thanks!