Metamath Proof Explorer


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 ( ( 𝜑𝜓 ) → cadd ( 𝜑 , 𝜓 , 𝜒 ) )

Proof

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