How to prove De Morgan’s laws without using contradictions

logic

$(A \land B) \to ¬(¬A \lor ¬B)$ and $A \lor B \to ¬(¬A \land ¬B)$ are intuitionistically valid.

Thus, we can prove them without Excluded Middle or other classically equivalent rule.

None of the converse implications is derivable in intuitionistic logic. Thus, we need LEM.

The proof of the first one is straightforward.

Assume $¬A \lor ¬B$ and derive a contradiction under both cases. Then conclude with $\lor$-elim followed by $\lnot$-intro.

The same for the second one, assuming $¬A \land ¬B$.

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