Theorem wl-3xornot1 35162

Description: Flipping the first input flips the triple xor. wl-3xorrot 35159 can rotate any input to the front, so flipping any one of them does the same. (Contributed by Wolf Lammen, 1-May-2024.)

Assertion

Ref	Expression
wl-3xornot1	⊢ (¬ hadd(𝜑, 𝜓, 𝜒) ↔ hadd(¬ 𝜑, 𝜓, 𝜒))

Proof of Theorem wl-3xornot1

Step	Hyp	Ref	Expression
1		wl-3xorbi 35155	. 2 ⊢ (hadd(¬ 𝜑, 𝜓, 𝜒) ↔ (¬ 𝜑 ↔ (𝜓 ↔ 𝜒)))
2		nbbn 389	. . 3 ⊢ ((¬ 𝜑 ↔ (𝜓 ↔ 𝜒)) ↔ ¬ (𝜑 ↔ (𝜓 ↔ 𝜒)))
3		wl-3xorbi 35155	. . 3 ⊢ (hadd(𝜑, 𝜓, 𝜒) ↔ (𝜑 ↔ (𝜓 ↔ 𝜒)))
4	2, 3	xchbinxr 339	. 2 ⊢ ((¬ 𝜑 ↔ (𝜓 ↔ 𝜒)) ↔ ¬ hadd(𝜑, 𝜓, 𝜒))
5	1, 4	bitr2i 279	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)