Definition df-cad 1609

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 1610 and cadan 1611 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 1608	. 2 wff cadd(𝜑, 𝜓, 𝜒)
5	1, 2	wa 396	. . 3 wff (𝜑 ∧ 𝜓)
6	1, 2	wxo 1506	. . . 4 wff (𝜑 ⊻ 𝜓)
7	3, 6	wa 396	. . 3 wff (𝜒 ∧ (𝜑 ⊻ 𝜓))
8	5, 7	wo 844	. 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  1610  cadbi123d  1612  cadcoma  1614  cad11  1618  cad0  1620  cad0OLD  1621