Theorem wl-3xornot1 37415

Description: Flipping the first input flips the triple xor. wl-3xorrot 37412 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 37408	. 2 ⊢ (hadd(¬ 𝜑, 𝜓, 𝜒) ↔ (¬ 𝜑 ↔ (𝜓 ↔ 𝜒)))
2		nbbn 383	. . 3 ⊢ ((¬ 𝜑 ↔ (𝜓 ↔ 𝜒)) ↔ ¬ (𝜑 ↔ (𝜓 ↔ 𝜒)))
3		wl-3xorbi 37408	. . 3 ⊢ (hadd(𝜑, 𝜓, 𝜒) ↔ (𝜑 ↔ (𝜓 ↔ 𝜒)))
4	2, 3	xchbinxr 335	. 2 ⊢ ((¬ 𝜑 ↔ (𝜓 ↔ 𝜒)) ↔ ¬ hadd(𝜑, 𝜓, 𝜒))
5	1, 4	bitr2i 276	1 ⊢ (¬ hadd(𝜑, 𝜓, 𝜒) ↔ hadd(¬ 𝜑, 𝜓, 𝜒))

Syntax hints:  ¬ wn 3   ↔ wb 206  haddwhad 1592

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 207  df-an 396  df-or 848  df-ifp 1063  df-xor 1511  df-tru 1542  df-had 1593

This theorem is referenced by: (None)