Theorem hadass 1626

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

Assertion

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

Proof of Theorem hadass

Step	Hyp	Ref	Expression
1		df-had 1623	. 2 ⊢ (hadd(𝜑, 𝜓, 𝜒) ↔ ((𝜑 ⊻ 𝜓) ⊻ 𝜒))
2		xorass 1544	. 2 ⊢ (((𝜑 ⊻ 𝜓) ⊻ 𝜒) ↔ (𝜑 ⊻ (𝜓 ⊻ 𝜒)))
3	1, 2	bitri 278	1 ⊢ (hadd(𝜑, 𝜓, 𝜒) ↔ (𝜑 ⊻ (𝜓 ⊻ 𝜒)))

This proof depends on syntax axioms:   ↔ wb 209   ⊻ wxo 1540  haddwhad 1622

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 210  df-xor 1541  df-had 1623

This theorem is used by:  hadcomb  1629