I am reading a proposition. The proposition is as follows:
Let $M$ and $\bar{M}$ be two manifold, with $\{\mu_i\}$ and $\{\bar{\mu}_i\}$ co-frame, respectively. And there is an diffeomorphism s.t. $\phi^*d\bar{\mu}=d\mu$. The Maurer-Cartan equations for $M$ and $\bar{M}$ is
$$ \mathrm{d}\mu^k=\sum_{i<j}C_{ij}^k(x)\mu^i \wedge \mu^j, $$ and
$$ \mathrm{d}\bar{\mu}^k=\sum_{i<j}\bar{C}_{ij}^k(x)\bar{\mu}^i \wedge \bar{\mu}^j, $$
then by $\phi^*\mathrm{d}\bar{\mu}=\mathrm{d}\phi^*\bar{\mu}$, they are equivalent if and only if $$ \bar{C}_{ij}^k(\phi(x))=C_{ij}^k(x). $$
My question is "How it is proved $\bar{C}_{ij}^k(\phi(x))=C_{ij}^k(x)$ iff two manifold are equivalent"?