Theorem cadnot

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

	Ref	Expression
Assertion	cadnot	⊢ ( ¬ cadd ( 𝜑 , 𝜓 , 𝜒 ) ↔ cadd ( ¬ 𝜑 , ¬ 𝜓 , ¬ 𝜒 ) )

Proof

Step	Hyp	Ref	Expression
1		ianor	⊢ ( ¬ ( 𝜑 ∧ 𝜓 ) ↔ ( ¬ 𝜑 ∨ ¬ 𝜓 ) )
2		ianor	⊢ ( ¬ ( 𝜑 ∧ 𝜒 ) ↔ ( ¬ 𝜑 ∨ ¬ 𝜒 ) )
3		ianor	⊢ ( ¬ ( 𝜓 ∧ 𝜒 ) ↔ ( ¬ 𝜓 ∨ ¬ 𝜒 ) )
4	1 2 3	3anbi123i	⊢ ( ( ¬ ( 𝜑 ∧ 𝜓 ) ∧ ¬ ( 𝜑 ∧ 𝜒 ) ∧ ¬ ( 𝜓 ∧ 𝜒 ) ) ↔ ( ( ¬ 𝜑 ∨ ¬ 𝜓 ) ∧ ( ¬ 𝜑 ∨ ¬ 𝜒 ) ∧ ( ¬ 𝜓 ∨ ¬ 𝜒 ) ) )
5		3ioran	⊢ ( ¬ ( ( 𝜑 ∧ 𝜓 ) ∨ ( 𝜑 ∧ 𝜒 ) ∨ ( 𝜓 ∧ 𝜒 ) ) ↔ ( ¬ ( 𝜑 ∧ 𝜓 ) ∧ ¬ ( 𝜑 ∧ 𝜒 ) ∧ ¬ ( 𝜓 ∧ 𝜒 ) ) )
6		cador	⊢ ( cadd ( 𝜑 , 𝜓 , 𝜒 ) ↔ ( ( 𝜑 ∧ 𝜓 ) ∨ ( 𝜑 ∧ 𝜒 ) ∨ ( 𝜓 ∧ 𝜒 ) ) )
7	5 6	xchnxbir	⊢ ( ¬ cadd ( 𝜑 , 𝜓 , 𝜒 ) ↔ ( ¬ ( 𝜑 ∧ 𝜓 ) ∧ ¬ ( 𝜑 ∧ 𝜒 ) ∧ ¬ ( 𝜓 ∧ 𝜒 ) ) )
8		cadan	⊢ ( cadd ( ¬ 𝜑 , ¬ 𝜓 , ¬ 𝜒 ) ↔ ( ( ¬ 𝜑 ∨ ¬ 𝜓 ) ∧ ( ¬ 𝜑 ∨ ¬ 𝜒 ) ∧ ( ¬ 𝜓 ∨ ¬ 𝜒 ) ) )
9	4 7 8	3bitr4i	⊢ ( ¬ cadd ( 𝜑 , 𝜓 , 𝜒 ) ↔ cadd ( ¬ 𝜑 , ¬ 𝜓 , ¬ 𝜒 ) )