Theorem hadnot

Description: The adder sum distributes over negation. (Contributed by Mario Carneiro, 4-Sep-2016) (Proof shortened by Wolf Lammen, 11-Jul-2020)

	Ref	Expression
Assertion	hadnot	⊢ ( ¬ hadd ( 𝜑 , 𝜓 , 𝜒 ) ↔ hadd ( ¬ 𝜑 , ¬ 𝜓 , ¬ 𝜒 ) )

Proof

Step	Hyp	Ref	Expression
1		notbi	⊢ ( ( 𝜑 ↔ 𝜓 ) ↔ ( ¬ 𝜑 ↔ ¬ 𝜓 ) )
2	1	bibi1i	⊢ ( ( ( 𝜑 ↔ 𝜓 ) ↔ ¬ 𝜒 ) ↔ ( ( ¬ 𝜑 ↔ ¬ 𝜓 ) ↔ ¬ 𝜒 ) )
3		xor3	⊢ ( ¬ ( ( 𝜑 ↔ 𝜓 ) ↔ 𝜒 ) ↔ ( ( 𝜑 ↔ 𝜓 ) ↔ ¬ 𝜒 ) )
4		hadbi	⊢ ( hadd ( 𝜑 , 𝜓 , 𝜒 ) ↔ ( ( 𝜑 ↔ 𝜓 ) ↔ 𝜒 ) )
5	3 4	xchnxbir	⊢ ( ¬ hadd ( 𝜑 , 𝜓 , 𝜒 ) ↔ ( ( 𝜑 ↔ 𝜓 ) ↔ ¬ 𝜒 ) )
6		hadbi	⊢ ( hadd ( ¬ 𝜑 , ¬ 𝜓 , ¬ 𝜒 ) ↔ ( ( ¬ 𝜑 ↔ ¬ 𝜓 ) ↔ ¬ 𝜒 ) )
7	2 5 6	3bitr4i	⊢ ( ¬ hadd ( 𝜑 , 𝜓 , 𝜒 ) ↔ hadd ( ¬ 𝜑 , ¬ 𝜓 , ¬ 𝜒 ) )