Given that:
$$ SD\bigg( (r, \langle r, s \rangle),(r, b) \bigg) < \mathrm{negl}(n)$$
where $SD$ stands for statistical distance, $r$ is random uniform in $\{0,1\}^n$, $s$ is random uniform in $S \subseteq \{0,1\}^n$ and $b$ is a uniformly distributed bit.
It seems intuitive that, given a collision resistant hash function $h$, it should also hold that:
$$ SD\bigg( \big(h(s), r, \langle r, s \rangle \oplus 0\big),\big(h(s), r, \langle r, s \rangle \oplus 1\big) \bigg) < \mathrm{negl}(n)$$
but I cannot seem to prove this. I've tried using the formal definition of $SD$, but I don't know how to handle the fact there are tuples so I haven't even reached a point where I could incorporate the first claim.
Is there a way to show this from the first claim? Or am I wrong and there's a way to refute this? (for context -- I'm trying to show that $\big(h(s), r, \langle r, s \rangle \oplus b\big)$ is a statistically hiding commitment scheme).
Thanks.