Is the rule of explosion (aka ex falso sequitur quodlibet) something that needs to be proved?

Question

On this and this pages, there are proofs presented for this rule, but what confuses me though is that I think we actually need to show that $\bot\vdash Q$ holds, not $(P\wedge\neg P)\vdash Q$ or $\vdash(P\wedge\neg P)\rightarrow Q$. I know that $P\wedge\neg P$ is an obvious contradiction and in the natural deduction system I have seen you introduce a $\bot$ from $P$ and $\neg P$ on two separate lines, but still I don't think this necessarily means that $\bot$ is equivalent to $P\wedge\neg P$!