Theorem wl-3xornot 35652

Description: Triple xor distributes over negation. Copy of hadnot 1604. (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 319	. . 3 ⊢ ((𝜑 ↔ 𝜓) ↔ (¬ 𝜑 ↔ ¬ 𝜓))
2	1	bibi1i 339	. 2 ⊢ (((𝜑 ↔ 𝜓) ↔ ¬ 𝜒) ↔ ((¬ 𝜑 ↔ ¬ 𝜓) ↔ ¬ 𝜒))
3		xor3 384	. . 3 ⊢ (¬ ((𝜑 ↔ 𝜓) ↔ 𝜒) ↔ ((𝜑 ↔ 𝜓) ↔ ¬ 𝜒))
4		wl-3xorbi2 35645	. . 3 ⊢ (hadd(𝜑, 𝜓, 𝜒) ↔ ((𝜑 ↔ 𝜓) ↔ 𝜒))
5	3, 4	xchnxbir 333	. 2 ⊢ (¬ hadd(𝜑, 𝜓, 𝜒) ↔ ((𝜑 ↔ 𝜓) ↔ ¬ 𝜒))
6		wl-3xorbi2 35645	. 2 ⊢ (hadd(¬ 𝜑, ¬ 𝜓, ¬ 𝜒) ↔ ((¬ 𝜑 ↔ ¬ 𝜓) ↔ ¬ 𝜒))
7	2, 5, 6	3bitr4i 303	1 ⊢ (¬ hadd(𝜑, 𝜓, 𝜒) ↔ hadd(¬ 𝜑, ¬ 𝜓, ¬ 𝜒))

Syntax hints:  ¬ wn 3   ↔ wb 205  haddwhad 1594

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 206  df-an 397  df-or 845  df-ifp 1061  df-xor 1507  df-tru 1542  df-had 1595

This theorem is referenced by: (None)