Theorem wl-hadcomb

Description: Commutative law for the adders sum. (Contributed by Mario Carneiro, 4-Sep-2016) Alternative definition. (Revised by Wolf Lammen, 24-Apr-2024)

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

Proof

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