Going from output of D/C to input of D/C
You won't get the discrete $y_d[m]$ by re-arranging the equations, but by sampling the $y_c(t)$ at integral multiples of $T$. When you sample the RHS in equation1 exactly at $t=mT$ instances, you get $y_d[m]$. You need to remember that $sinc(m-n) = 1$ for $m=n$ and $0$ otherwise, and (BTW in equation 1, in $sinc$ argument $\pi$ won't be there). Anyway, take RHS and evaluate the expression at $t = mT$ where $m$ are integers.
$$y_c(t) \big|_{t=mT} = \sum^{\infty}_{n=-\infty}y_d[n] \, \mbox{sinc} \left( \frac{t-nT}{T} \right) \Big|_{t=mT} = y_d[n]$$