Theorem wl-3xorcoma 35160

Description: Commutative law for triple xor. Copy of hadcoma 1601. (Contributed by Mario Carneiro, 4-Sep-2016.) (Proof shortened by Wolf Lammen, 17-Dec-2023.)

Assertion

Ref	Expression
wl-3xorcoma	⊢ (hadd(𝜑, 𝜓, 𝜒) ↔ hadd(𝜓, 𝜑, 𝜒))

Proof of Theorem wl-3xorcoma

Step	Hyp	Ref	Expression
1		bicom 225	. . 3 ⊢ ((𝜑 ↔ 𝜓) ↔ (𝜓 ↔ 𝜑))
2	1	bibi1i 343	. 2 ⊢ (((𝜑 ↔ 𝜓) ↔ 𝜒) ↔ ((𝜓 ↔ 𝜑) ↔ 𝜒))
3		wl-3xorbi2 35156	. 2 ⊢ (hadd(𝜑, 𝜓, 𝜒) ↔ ((𝜑 ↔ 𝜓) ↔ 𝜒))
4		wl-3xorbi2 35156	. 2 ⊢ (hadd(𝜓, 𝜑, 𝜒) ↔ ((𝜓 ↔ 𝜑) ↔ 𝜒))
5	2, 3, 4	3bitr4i 307	1 ⊢ (hadd(𝜑, 𝜓, 𝜒) ↔ hadd(𝜓, 𝜑, 𝜒))

Syntax hints:   ↔ wb 209  haddwhad 1595

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 210  df-an 401  df-or 846  df-ifp 1060  df-xor 1504  df-tru 1542  df-had 1596

This theorem is referenced by:  wl-3xorcomb  35161