Definition df-cad 1604

Description: Definition of the "carry" output of the full adder. It is true when at least two arguments are true, so it is equal to the "majority" function on three variables. See cador 1605 and cadan 1606 for alternate definitions. (Contributed by Mario Carneiro, 4-Sep-2016.)

Assertion

Ref	Expression
df-cad	⊢ (cadd(𝜑, 𝜓, 𝜒) ↔ ((𝜑 ∧ 𝜓) ∨ (𝜒 ∧ (𝜑 ⊻ 𝜓))))

Detailed syntax breakdown of Definition df-cad

Step	Hyp	Ref	Expression
1		wph	. . 3 wff 𝜑
2		wps	. . 3 wff 𝜓
3		wch	. . 3 wff 𝜒
4	1, 2, 3	wcad 1603	. 2 wff cadd(𝜑, 𝜓, 𝜒)
5	1, 2	wa 395	. . 3 wff (𝜑 ∧ 𝜓)
6	1, 2	wxo 1508	. . . 4 wff (𝜑 ⊻ 𝜓)
7	3, 6	wa 395	. . 3 wff (𝜒 ∧ (𝜑 ⊻ 𝜓))
8	5, 7	wo 846	. 2 wff ((𝜑 ∧ 𝜓) ∨ (𝜒 ∧ (𝜑 ⊻ 𝜓)))
9	4, 8	wb 206	1 wff (cadd(𝜑, 𝜓, 𝜒) ↔ ((𝜑 ∧ 𝜓) ∨ (𝜒 ∧ (𝜑 ⊻ 𝜓))))

This definition is referenced by:  cador  1605  cadbi123d  1607  cadcoma  1609  cad11  1613  cad0  1615  cad0OLD  1616