Theorem hadnot

Description: The adder sum distributes over negation. (Contributed by Mario Carneiro, 4-Sep-2016) (Proof shortened by Wolf Lammen, 11-Jul-2020)

	Ref	Expression
Assertion	hadnot	$$⊢\neg hadd\left(\phi ,\psi ,\chi \right)\leftrightarrow hadd\left(\neg \phi ,\neg \psi ,\neg \chi \right)$$

Proof

Step	Hyp	Ref	Expression
1		notbi	$$⊢\left(\phi \leftrightarrow \psi \right)\leftrightarrow \left(\neg \phi \leftrightarrow \neg \psi \right)$$
2	1	bibi1i	$$⊢\left(\left(\phi \leftrightarrow \psi \right)\leftrightarrow \neg \chi \right)\leftrightarrow \left(\left(\neg \phi \leftrightarrow \neg \psi \right)\leftrightarrow \neg \chi \right)$$
3		xor3	$$⊢\neg \left(\left(\phi \leftrightarrow \psi \right)\leftrightarrow \chi \right)\leftrightarrow \left(\left(\phi \leftrightarrow \psi \right)\leftrightarrow \neg \chi \right)$$
4		hadbi	$$⊢hadd\left(\phi ,\psi ,\chi \right)\leftrightarrow \left(\left(\phi \leftrightarrow \psi \right)\leftrightarrow \chi \right)$$
5	3 4	xchnxbir	$$⊢\neg hadd\left(\phi ,\psi ,\chi \right)\leftrightarrow \left(\left(\phi \leftrightarrow \psi \right)\leftrightarrow \neg \chi \right)$$
6		hadbi	$$⊢hadd\left(\neg \phi ,\neg \psi ,\neg \chi \right)\leftrightarrow \left(\left(\neg \phi \leftrightarrow \neg \psi \right)\leftrightarrow \neg \chi \right)$$
7	2 5 6	3bitr4i	$$⊢\neg hadd\left(\phi ,\psi ,\chi \right)\leftrightarrow hadd\left(\neg \phi ,\neg \psi ,\neg \chi \right)$$