Theorem wl-3xornot 35163

Description: Triple xor distributes over negation. Copy of hadnot 1605. (Contributed by Mario Carneiro, 4-Sep-2016.) (Proof shortened by Wolf Lammen, 11-Jul-2020.)

Assertion

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

Proof of Theorem wl-3xornot

Step	Hyp	Ref	Expression
1		notbi 323	. . 3 ⊢ ((𝜑 ↔ 𝜓) ↔ (¬ 𝜑 ↔ ¬ 𝜓))
2	1	bibi1i 343	. 2 ⊢ (((𝜑 ↔ 𝜓) ↔ ¬ 𝜒) ↔ ((¬ 𝜑 ↔ ¬ 𝜓) ↔ ¬ 𝜒))
3		xor3 388	. . 3 ⊢ (¬ ((𝜑 ↔ 𝜓) ↔ 𝜒) ↔ ((𝜑 ↔ 𝜓) ↔ ¬ 𝜒))
4		wl-3xorbi2 35156	. . 3 ⊢ (hadd(𝜑, 𝜓, 𝜒) ↔ ((𝜑 ↔ 𝜓) ↔ 𝜒))
5	3, 4	xchnxbir 337	. 2 ⊢ (¬ hadd(𝜑, 𝜓, 𝜒) ↔ ((𝜑 ↔ 𝜓) ↔ ¬ 𝜒))
6		wl-3xorbi2 35156	. 2 ⊢ (hadd(¬ 𝜑, ¬ 𝜓, ¬ 𝜒) ↔ ((¬ 𝜑 ↔ ¬ 𝜓) ↔ ¬ 𝜒))
7	2, 5, 6	3bitr4i 307	1 ⊢ (¬ hadd(𝜑, 𝜓, 𝜒) ↔ hadd(¬ 𝜑, ¬ 𝜓, ¬ 𝜒))

Syntax hints:  ¬ wn 3   ↔ 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: (None)