Theorem cador

Description: The adder carry in disjunctive normal form. (Contributed by Mario Carneiro, 4-Sep-2016) (Proof shortened by Wolf Lammen, 11-Jul-2020)

	Ref	Expression
Assertion	cador	⊢ ( cadd ( 𝜑 , 𝜓 , 𝜒 ) ↔ ( ( 𝜑 ∧ 𝜓 ) ∨ ( 𝜑 ∧ 𝜒 ) ∨ ( 𝜓 ∧ 𝜒 ) ) )

Proof

Step	Hyp	Ref	Expression
1		xor2	⊢ ( ( 𝜑 ⊻ 𝜓 ) ↔ ( ( 𝜑 ∨ 𝜓 ) ∧ ¬ ( 𝜑 ∧ 𝜓 ) ) )
2	1	rbaib	⊢ ( ¬ ( 𝜑 ∧ 𝜓 ) → ( ( 𝜑 ⊻ 𝜓 ) ↔ ( 𝜑 ∨ 𝜓 ) ) )
3	2	anbi1d	⊢ ( ¬ ( 𝜑 ∧ 𝜓 ) → ( ( ( 𝜑 ⊻ 𝜓 ) ∧ 𝜒 ) ↔ ( ( 𝜑 ∨ 𝜓 ) ∧ 𝜒 ) ) )
4		ancom	⊢ ( ( ( 𝜑 ⊻ 𝜓 ) ∧ 𝜒 ) ↔ ( 𝜒 ∧ ( 𝜑 ⊻ 𝜓 ) ) )
5		andir	⊢ ( ( ( 𝜑 ∨ 𝜓 ) ∧ 𝜒 ) ↔ ( ( 𝜑 ∧ 𝜒 ) ∨ ( 𝜓 ∧ 𝜒 ) ) )
6	3 4 5	3bitr3g	⊢ ( ¬ ( 𝜑 ∧ 𝜓 ) → ( ( 𝜒 ∧ ( 𝜑 ⊻ 𝜓 ) ) ↔ ( ( 𝜑 ∧ 𝜒 ) ∨ ( 𝜓 ∧ 𝜒 ) ) ) )
7	6	pm5.74i	⊢ ( ( ¬ ( 𝜑 ∧ 𝜓 ) → ( 𝜒 ∧ ( 𝜑 ⊻ 𝜓 ) ) ) ↔ ( ¬ ( 𝜑 ∧ 𝜓 ) → ( ( 𝜑 ∧ 𝜒 ) ∨ ( 𝜓 ∧ 𝜒 ) ) ) )
8		df-or	⊢ ( ( ( 𝜑 ∧ 𝜓 ) ∨ ( 𝜒 ∧ ( 𝜑 ⊻ 𝜓 ) ) ) ↔ ( ¬ ( 𝜑 ∧ 𝜓 ) → ( 𝜒 ∧ ( 𝜑 ⊻ 𝜓 ) ) ) )
9		df-or	⊢ ( ( ( 𝜑 ∧ 𝜓 ) ∨ ( ( 𝜑 ∧ 𝜒 ) ∨ ( 𝜓 ∧ 𝜒 ) ) ) ↔ ( ¬ ( 𝜑 ∧ 𝜓 ) → ( ( 𝜑 ∧ 𝜒 ) ∨ ( 𝜓 ∧ 𝜒 ) ) ) )
10	7 8 9	3bitr4i	⊢ ( ( ( 𝜑 ∧ 𝜓 ) ∨ ( 𝜒 ∧ ( 𝜑 ⊻ 𝜓 ) ) ) ↔ ( ( 𝜑 ∧ 𝜓 ) ∨ ( ( 𝜑 ∧ 𝜒 ) ∨ ( 𝜓 ∧ 𝜒 ) ) ) )
11		df-cad	⊢ ( cadd ( 𝜑 , 𝜓 , 𝜒 ) ↔ ( ( 𝜑 ∧ 𝜓 ) ∨ ( 𝜒 ∧ ( 𝜑 ⊻ 𝜓 ) ) ) )
12		3orass	⊢ ( ( ( 𝜑 ∧ 𝜓 ) ∨ ( 𝜑 ∧ 𝜒 ) ∨ ( 𝜓 ∧ 𝜒 ) ) ↔ ( ( 𝜑 ∧ 𝜓 ) ∨ ( ( 𝜑 ∧ 𝜒 ) ∨ ( 𝜓 ∧ 𝜒 ) ) ) )
13	10 11 12	3bitr4i	⊢ ( cadd ( 𝜑 , 𝜓 , 𝜒 ) ↔ ( ( 𝜑 ∧ 𝜓 ) ∨ ( 𝜑 ∧ 𝜒 ) ∨ ( 𝜓 ∧ 𝜒 ) ) )