Theorem cad1

Description: If one input is true, then the adder carry is true exactly when at least one of the other two inputs is true. (Contributed by Mario Carneiro, 8-Sep-2016) (Proof shortened by Wolf Lammen, 19-Jun-2020)

	Ref	Expression
Assertion	cad1	|- ( ch -> ( cadd ( ph , ps , ch ) <-> ( ph \/ ps ) ) )

Proof

Step	Hyp	Ref	Expression
1		cadan	|- ( cadd ( ph , ps , ch ) <-> ( ( ph \/ ps ) /\ ( ph \/ ch ) /\ ( ps \/ ch ) ) )
2		3anass	|- ( ( ( ph \/ ps ) /\ ( ph \/ ch ) /\ ( ps \/ ch ) ) <-> ( ( ph \/ ps ) /\ ( ( ph \/ ch ) /\ ( ps \/ ch ) ) ) )
3	1 2	bitri	|- ( cadd ( ph , ps , ch ) <-> ( ( ph \/ ps ) /\ ( ( ph \/ ch ) /\ ( ps \/ ch ) ) ) )
4		olc	|- ( ch -> ( ph \/ ch ) )
5		olc	|- ( ch -> ( ps \/ ch ) )
6	4 5	jca	|- ( ch -> ( ( ph \/ ch ) /\ ( ps \/ ch ) ) )
7	6	biantrud	|- ( ch -> ( ( ph \/ ps ) <-> ( ( ph \/ ps ) /\ ( ( ph \/ ch ) /\ ( ps \/ ch ) ) ) ) )
8	3 7	bitr4id	|- ( ch -> ( cadd ( ph , ps , ch ) <-> ( ph \/ ps ) ) )