Theorem cad11

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

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

Proof

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