Theorem hadcoma 1538

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

Assertion

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

Proof of Theorem hadcoma

Step	Hyp	Ref	Expression
1		xorcom 1467	. . 3 ⊢ ((𝜑 ⊻ 𝜓) ↔ (𝜓 ⊻ 𝜑))
2		biid 251	. . 3 ⊢ (𝜒 ↔ 𝜒)
3	1, 2	xorbi12i 1477	. 2 ⊢ (((𝜑 ⊻ 𝜓) ⊻ 𝜒) ↔ ((𝜓 ⊻ 𝜑) ⊻ 𝜒))
4		df-had 1533	. 2 ⊢ (hadd(𝜑, 𝜓, 𝜒) ↔ ((𝜑 ⊻ 𝜓) ⊻ 𝜒))
5		df-had 1533	. 2 ⊢ (hadd(𝜓, 𝜑, 𝜒) ↔ ((𝜓 ⊻ 𝜑) ⊻ 𝜒))
6	3, 4, 5	3bitr4i 292	1 ⊢ (hadd(𝜑, 𝜓, 𝜒) ↔ hadd(𝜓, 𝜑, 𝜒))

Syntax hints:   ↔ wb 196   ⊻ wxo 1464  haddwhad 1532

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 197  df-xor 1465  df-had 1533

This theorem is referenced by:  hadrot  1540  sadcom  15185