Theorem hadcomb 1389

Description: Commutative law for triple XOR. (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 227	. . 3 ⊢ (φ ↔ φ)
2		xorcom 1307	. . 3 ⊢ ((ψ ⊻ χ) ↔ (χ ⊻ ψ))
3	1, 2	xorbi12i 1314	. 2 ⊢ ((φ ⊻ (ψ ⊻ χ)) ↔ (φ ⊻ (χ ⊻ ψ)))
4		hadass 1386	. 2 ⊢ (hadd(φ, ψ, χ) ↔ (φ ⊻ (ψ ⊻ χ)))
5		hadass 1386	. 2 ⊢ (hadd(φ, χ, ψ) ↔ (φ ⊻ (χ ⊻ ψ)))
6	3, 4, 5	3bitr4i 268	1 ⊢ (hadd(φ, ψ, χ) ↔ hadd(φ, χ, ψ))

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