Theorem hadcomb 1605

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

Assertion

Ref	Expression
hadcomb	⊢ (hadd(𝜑, 𝜓, 𝜒) ↔ hadd(𝜑, 𝜒, 𝜓))

Proof of Theorem hadcomb

Step	Hyp	Ref	Expression
1		biid 260	. . 3 ⊢ (𝜑 ↔ 𝜑)
2		xorcom 1508	. . 3 ⊢ ((𝜓 ⊻ 𝜒) ↔ (𝜒 ⊻ 𝜓))
3	1, 2	xorbi12i 1519	. 2 ⊢ ((𝜑 ⊻ (𝜓 ⊻ 𝜒)) ↔ (𝜑 ⊻ (𝜒 ⊻ 𝜓)))
4		hadass 1601	. 2 ⊢ (hadd(𝜑, 𝜓, 𝜒) ↔ (𝜑 ⊻ (𝜓 ⊻ 𝜒)))
5		hadass 1601	. 2 ⊢ (hadd(𝜑, 𝜒, 𝜓) ↔ (𝜑 ⊻ (𝜒 ⊻ 𝜓)))
6	3, 4, 5	3bitr4i 302	1 ⊢ (hadd(𝜑, 𝜓, 𝜒) ↔ hadd(𝜑, 𝜒, 𝜓))

Syntax hints:   ↔ wb 205   ⊻ wxo 1505  haddwhad 1597

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 206  df-xor 1506  df-had 1598

This theorem is referenced by:  hadrot  1606