Theorem hadass

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

	Ref	Expression
Assertion	hadass	⊢ ( hadd ( 𝜑 , 𝜓 , 𝜒 ) ↔ ( 𝜑 ⊻ ( 𝜓 ⊻ 𝜒 ) ) )

Proof

Step	Hyp	Ref	Expression
1		df-had	⊢ ( hadd ( 𝜑 , 𝜓 , 𝜒 ) ↔ ( ( 𝜑 ⊻ 𝜓 ) ⊻ 𝜒 ) )
2		xorass	⊢ ( ( ( 𝜑 ⊻ 𝜓 ) ⊻ 𝜒 ) ↔ ( 𝜑 ⊻ ( 𝜓 ⊻ 𝜒 ) ) )
3	1 2	bitri	⊢ ( hadd ( 𝜑 , 𝜓 , 𝜒 ) ↔ ( 𝜑 ⊻ ( 𝜓 ⊻ 𝜒 ) ) )