Tautology is Negation of Contradiction/Proof 3

Proof
Let $p$ be a propositional formula.

Let $v$ be an arbitrary boolean interpretation of $p$.

Then:


 * $\map v p = T \iff \map v {\neg p} = F$

by the definition of the logical not.

Since $v$ is arbitrary, $p$ is true in all interpretations $\neg p$ is false in all interpretations.

Hence:
 * $\top \dashv \vdash \neg \bot$