Theorem wl-3xorcomb

Description: Commutative 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-3xorcomb	$$⊢hadd\left(\phi ,\psi ,\chi \right)\leftrightarrow hadd\left(\phi ,\chi ,\psi \right)$$

Proof

Step	Hyp	Ref	Expression
1		wl-3xorcoma	$$⊢hadd\left(\phi ,\psi ,\chi \right)\leftrightarrow hadd\left(\psi ,\phi ,\chi \right)$$
2		wl-3xorrot	$$⊢hadd\left(\psi ,\phi ,\chi \right)\leftrightarrow hadd\left(\phi ,\chi ,\psi \right)$$
3	1 2	bitri	$$⊢hadd\left(\phi ,\psi ,\chi \right)\leftrightarrow hadd\left(\phi ,\chi ,\psi \right)$$