I am asking the question not purely for a logic exercise but I am just trying to prove something by contrapositive and this got me confused.

So when $A$ is a statement where, say, three conditions need to hold e.g. $P \wedge Q \wedge R$ and $B := S$, proving the contrapositive would be showing $\neg S \Rightarrow \neg P \lor \neg Q \lor \neg R$. At first I assumed $S$ does not hold **AND** $A$ and $B$ do hold to show that this contradicts $C$ but this is apparently wrong. Now, if I want to prove the contrapositive is the only way by showing that $\neg S$ leads to a contradiction with one of the statements $A,B$ or $C$? This also looks not quite right because if it were the case then surely then that would mean only one of those statements $A, B$ or $C$ imply $S$.