Tautology is Negation of Contradiction/Proof 3

Theorem
A tautology implies and is implied by the negation of a contradiction:


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

That is, a truth can not be false, and a non-falsehood must be a truth.

Proof
Let $p$ be a propositional formula.

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

Then:


 * $v \left({p}\right) = T \iff v \left({\neg p}\right) = F$

by the definition of the logical not.

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

Hence $\top \dashv \vdash \neg \bot$.