Theorem wl-3xornot1

Description: Flipping the first input flips the triple xor. wl-3xorrot can rotate any input to the front, so flipping any one of them does the same. (Contributed by Wolf Lammen, 1-May-2024)

	Ref	Expression
Assertion	wl-3xornot1	$$⊢\neg hadd\left(\phi ,\psi ,\chi \right)\leftrightarrow hadd\left(\neg \phi ,\psi ,\chi \right)$$

Proof

Step	Hyp	Ref	Expression
1		wl-3xorbi	$$⊢hadd\left(\neg \phi ,\psi ,\chi \right)\leftrightarrow \left(\neg \phi \leftrightarrow \left(\psi \leftrightarrow \chi \right)\right)$$
2		nbbn	$$⊢\left(\neg \phi \leftrightarrow \left(\psi \leftrightarrow \chi \right)\right)\leftrightarrow \neg \left(\phi \leftrightarrow \left(\psi \leftrightarrow \chi \right)\right)$$
3		wl-3xorbi	$$⊢hadd\left(\phi ,\psi ,\chi \right)\leftrightarrow \left(\phi \leftrightarrow \left(\psi \leftrightarrow \chi \right)\right)$$
4	2 3	xchbinxr	$$⊢\left(\neg \phi \leftrightarrow \left(\psi \leftrightarrow \chi \right)\right)\leftrightarrow \neg hadd\left(\phi ,\psi ,\chi \right)$$
5	1 4	bitr2i	$$⊢\neg hadd\left(\phi ,\psi ,\chi \right)\leftrightarrow hadd\left(\neg \phi ,\psi ,\chi \right)$$