Theorem cad11 1619

Description: If (at least) two inputs are true, then the adder carry is true. (Contributed by Mario Carneiro, 4-Sep-2016.)

Assertion

Ref	Expression
cad11	⊢ ((𝜑 ∧ 𝜓) → cadd(𝜑, 𝜓, 𝜒))

Proof of Theorem cad11

Step	Hyp	Ref	Expression
1		orc 863	. 2 ⊢ ((𝜑 ∧ 𝜓) → ((𝜑 ∧ 𝜓) ∨ (𝜒 ∧ (𝜑 ⊻ 𝜓))))
2		df-cad 1610	. 2 ⊢ (cadd(𝜑, 𝜓, 𝜒) ↔ ((𝜑 ∧ 𝜓) ∨ (𝜒 ∧ (𝜑 ⊻ 𝜓))))
3	1, 2	sylibr 233	1 ⊢ ((𝜑 ∧ 𝜓) → cadd(𝜑, 𝜓, 𝜒))

Syntax hints:   → wi 4   ∧ wa 395   ∨ wo 843   ⊻ wxo 1503  caddwcad 1609

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 206  df-or 844  df-cad 1610

This theorem is referenced by:  cad0  1621  cadtru  1624