Theorem wl-3xorrot

Description: Rotation law for triple xor. (Contributed by Mario Carneiro, 4-Sep-2016) df-had redefined. (Revised by Wolf Lammen, 24-Apr-2024)

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

Proof

Step	Hyp	Ref	Expression
1		bicom	$$⊢\left(\phi \leftrightarrow \left(\psi \leftrightarrow \chi \right)\right)\leftrightarrow \left(\left(\psi \leftrightarrow \chi \right)\leftrightarrow \phi \right)$$
2		wl-3xorbi	$$⊢hadd\left(\phi ,\psi ,\chi \right)\leftrightarrow \left(\phi \leftrightarrow \left(\psi \leftrightarrow \chi \right)\right)$$
3		wl-3xorbi2	$$⊢hadd\left(\psi ,\chi ,\phi \right)\leftrightarrow \left(\left(\psi \leftrightarrow \chi \right)\leftrightarrow \phi \right)$$
4	1 2 3	3bitr4i	$$⊢hadd\left(\phi ,\psi ,\chi \right)\leftrightarrow hadd\left(\psi ,\chi ,\phi \right)$$