Theorem wl-3xornot1

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

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

Proof

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