Theorem hadass 1386

Description: Associative law for triple XOR. (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 1380	. 2 ⊢ (hadd(φ, ψ, χ) ↔ ((φ ⊻ ψ) ⊻ χ))
2		xorass 1308	. 2 ⊢ (((φ ⊻ ψ) ⊻ χ) ↔ (φ ⊻ (ψ ⊻ χ)))
3	1, 2	bitri 240	1 ⊢ (hadd(φ, ψ, χ) ↔ (φ ⊻ (ψ ⊻ 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:  hadcomb  1389