Definition df-cad 1610

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 1611 and cadan 1612 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 1609	. 2 wff cadd(𝜑, 𝜓, 𝜒)
5	1, 2	wa 395	. . 3 wff (𝜑 ∧ 𝜓)
6	1, 2	wxo 1503	. . . 4 wff (𝜑 ⊻ 𝜓)
7	3, 6	wa 395	. . 3 wff (𝜒 ∧ (𝜑 ⊻ 𝜓))
8	5, 7	wo 843	. 2 wff ((𝜑 ∧ 𝜓) ∨ (𝜒 ∧ (𝜑 ⊻ 𝜓)))
9	4, 8	wb 205	1 wff (cadd(𝜑, 𝜓, 𝜒) ↔ ((𝜑 ∧ 𝜓) ∨ (𝜒 ∧ (𝜑 ⊻ 𝜓))))

This definition is referenced by:  cador  1611  cadbi123d  1613  cadcoma  1615  cad11  1619  cad0  1621  cad0OLD  1622