Theorem hadass

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

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

Proof

Step	Hyp	Ref	Expression
1		df-had	$$⊢hadd\left(\phi ,\psi ,\chi \right)\leftrightarrow \left(\left(\phi ⊻\psi \right)⊻\chi \right)$$
2		xorass	$$⊢\left(\left(\phi ⊻\psi \right)⊻\chi \right)\leftrightarrow \left(\phi ⊻\left(\psi ⊻\chi \right)\right)$$
3	1 2	bitri	$$⊢hadd\left(\phi ,\psi ,\chi \right)\leftrightarrow \left(\phi ⊻\left(\psi ⊻\chi \right)\right)$$