Let $I'\subset B\otimes_A B$ be the ideal generated by the elements $b_i\otimes 1-1\otimes b_i$, and define
$$
R=\{b\in B:b\otimes 1-1\otimes b\in I'\}.
$$
It’s not hard to check that $R$ is an $A$-subalgebra of $B$, so that $R=B$ (because the generators $b_i$ are in $R$ by construction). Now $b\otimes 1-1\otimes b\in I'$ for all $b\in B$ implies $b\otimes b'-bb'\otimes 1 = (b\otimes 1)(1\otimes b'-b'\otimes  1)\in I'$, so that $s-\nabla(s)\otimes 1\in I'$ for all $s\in B\otimes_A B$, where $\nabla:B\otimes B\to B$ is the codiagonal (i.e., the linear map sending every $b \otimes b'$ to $bb'$). Finally, if $s\in\ker(\nabla)$, then $s-\nabla(s)\otimes 1=s\in I’$.