Theorem hadcoma 1388

Description: Commutative law for triple XOR. (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 1307	. . 3 ⊢ ((φ ⊻ ψ) ↔ (ψ ⊻ φ))
2		biid 227	. . 3 ⊢ (χ ↔ χ)
3	1, 2	xorbi12i 1314	. 2 ⊢ (((φ ⊻ ψ) ⊻ χ) ↔ ((ψ ⊻ φ) ⊻ χ))
4		df-had 1380	. 2 ⊢ (hadd(φ, ψ, χ) ↔ ((φ ⊻ ψ) ⊻ χ))
5		df-had 1380	. 2 ⊢ (hadd(ψ, φ, χ) ↔ ((ψ ⊻ φ) ⊻ χ))
6	3, 4, 5	3bitr4i 268	1 ⊢ (hadd(φ, ψ, χ) ↔ hadd(ψ, φ, χ))

Syntax hints:   ↔ wb 176   ⊻ wxo 1304  haddwhad 1378

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 177  df-xor 1305  df-had 1380

This theorem is referenced by:  hadrot  1390