Theorem hadcomb

Description: Commutative law for the adders sum. (Contributed by Mario Carneiro, 4-Sep-2016)

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

Proof

Step	Hyp	Ref	Expression
1		biid	$$⊢\phi \leftrightarrow \phi$$
2		xorcom	$$⊢\left(\psi ⊻\chi \right)\leftrightarrow \left(\chi ⊻\psi \right)$$
3	1 2	xorbi12i	$$⊢\left(\phi ⊻\left(\psi ⊻\chi \right)\right)\leftrightarrow \left(\phi ⊻\left(\chi ⊻\psi \right)\right)$$
4		hadass	$$⊢hadd\left(\phi ,\psi ,\chi \right)\leftrightarrow \left(\phi ⊻\left(\psi ⊻\chi \right)\right)$$
5		hadass	$$⊢hadd\left(\phi ,\chi ,\psi \right)\leftrightarrow \left(\phi ⊻\left(\chi ⊻\psi \right)\right)$$
6	3 4 5	3bitr4i	$$⊢hadd\left(\phi ,\psi ,\chi \right)\leftrightarrow hadd\left(\phi ,\chi ,\psi \right)$$