Intuitionistic logic

The principle of non-contradiction is equivalent to ${\displaystyle \neg {\big (}(\neg A\to A)\land \neg A{\big )}}$, which in turn is equivalent to

${\displaystyle (\neg A\to A)\to \neg \neg A}$

This result can also be seen as the special case of

${\displaystyle {\big (}(A\to B)\to A{\big )}\to {\big (}(A\to B)\to B{\big )}}$

for ${\displaystyle B=\bot }$. This proposition follows from the propositional form of modus ponens ${\displaystyle A\to {\big (}(A\to B)\to B{\big )}}$ together with the fact that always ${\displaystyle A\to (C\to B)\vdash (C\to A)\to (C\to B)}$.

Since ${\displaystyle D\to \neg \neg A}$ is always also intuitionistically equivalent to ${\displaystyle \neg \neg (D\to A)}$, the above also constructively establishes the double negation of consequentia mirabilis.

Classical logic

Consequentia mirabilis follows from the above by double-negation elimination.

Indeed, the result is also established when using the classically valid propositional form of the reverse disjunctive syllogism ${\displaystyle (B\to A)\to (\neg B\lor A)}$ with the double-negation elimination principle in the form ${\displaystyle (\neg \neg A\lor A)\to A}$.

Related to the second intuitionistic derivation given above, consequentia mirabilis also follow as the special case of Pierce's law

${\displaystyle {\big (}(A\to B)\to A{\big )}\to A}$

for ${\displaystyle B=\bot }$.