Theorem cad0

Description: If one input is false, then the adder carry is true exactly when both of the other two inputs are true. (Contributed by Mario Carneiro, 8-Sep-2016) (Proof shortened by Wolf Lammen, 21-Sep-2024)

	Ref	Expression
Assertion	cad0	|- ( -. ch -> ( cadd ( ph , ps , ch ) <-> ( ph /\ ps ) ) )

Proof

Step	Hyp	Ref	Expression
1		df-cad	|- ( cadd ( ph , ps , ch ) <-> ( ( ph /\ ps ) \/ ( ch /\ ( ph \/_ ps ) ) ) )
2		idd	|- ( -. ch -> ( ( ph /\ ps ) -> ( ph /\ ps ) ) )
3		pm2.21	|- ( -. ch -> ( ch -> ( ph /\ ps ) ) )
4	3	adantrd	|- ( -. ch -> ( ( ch /\ ( ph \/_ ps ) ) -> ( ph /\ ps ) ) )
5	2 4	jaod	|- ( -. ch -> ( ( ( ph /\ ps ) \/ ( ch /\ ( ph \/_ ps ) ) ) -> ( ph /\ ps ) ) )
6	1 5	biimtrid	|- ( -. ch -> ( cadd ( ph , ps , ch ) -> ( ph /\ ps ) ) )
7		cad11	|- ( ( ph /\ ps ) -> cadd ( ph , ps , ch ) )
8	6 7	impbid1	|- ( -. ch -> ( cadd ( ph , ps , ch ) <-> ( ph /\ ps ) ) )