Definition df-cad 1637

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 1638 and cadan 1639 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 1636	. 2 wff cadd(𝜑, 𝜓, 𝜒)
5	1, 2	wa 400	. . 3 wff (𝜑 ∧ 𝜓)
6	1, 2	wxo 1541	. . . 4 wff (𝜑 ⊻ 𝜓)
7	3, 6	wa 400	. . 3 wff (𝜒 ∧ (𝜑 ⊻ 𝜓))
8	5, 7	wo 860	. 2 wff ((𝜑 ∧ 𝜓) ∨ (𝜒 ∧ (𝜑 ⊻ 𝜓)))
9	4, 8	wb 209	1 wff (cadd(𝜑, 𝜓, 𝜒) ↔ ((𝜑 ∧ 𝜓) ∨ (𝜒 ∧ (𝜑 ⊻ 𝜓))))

This definition is referenced by:  cador  1638  cadbi123d  1640  cadcoma  1642  cad11  1646  cad0  1648