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	$$⊢\left(\phi \wedge \psi \right)\to cadd\left(\phi ,\psi ,\chi \right)$$

Proof

Step	Hyp	Ref	Expression
1		orc	$$⊢\left(\phi \wedge \psi \right)\to \left(\left(\phi \wedge \psi \right)\vee \left(\chi \wedge \left(\phi ⊻\psi \right)\right)\right)$$
2		df-cad	$$⊢cadd\left(\phi ,\psi ,\chi \right)\leftrightarrow \left(\left(\phi \wedge \psi \right)\vee \left(\chi \wedge \left(\phi ⊻\psi \right)\right)\right)$$
3	1 2	sylibr	$$⊢\left(\phi \wedge \psi \right)\to cadd\left(\phi ,\psi ,\chi \right)$$