Note that $b \Rightarrow (\neg a)$ is the same as $(\neg b) \vee \neg (a)$. Moreover, this is the same as $\neg (a \wedge b)$.
Let $P$ abbreviate $a \wedge b$. In particular 2. is $P$. From the first paragraph, we have shown that 1. gives $\neg P$.
Hence you have $P \wedge \neg P$.
By using truth table, you can check that $(P \wedge \neg P) \Rightarrow c$ is a Tautology (always true) for all $c$.
So far, we have $P \wedge \neg P$ and $(P \wedge P) \Rightarrow c$. By Modus Ponen, you have $c$.
A formal system is sometimes called inconsistent if it can prove a contradiction (i.e. $P \wedge \neg P$). An alternative definition of inconsistent is that the formal system can prove everything. The argument here shows that the two definition are equivalent.