Theorem cadbi123d

Description: Equality theorem for the adder carry. (Contributed by Mario Carneiro, 4-Sep-2016)

	Ref	Expression
Hypotheses	cadbid.1	|- ( ph -> ( ps <-> ch ) )
	cadbid.2	|- ( ph -> ( th <-> ta ) )
	cadbid.3	|- ( ph -> ( et <-> ze ) )
Assertion	cadbi123d	|- ( ph -> ( cadd ( ps , th , et ) <-> cadd ( ch , ta , ze ) ) )

Proof

Step	Hyp	Ref	Expression
1		cadbid.1	|- ( ph -> ( ps <-> ch ) )
2		cadbid.2	|- ( ph -> ( th <-> ta ) )
3		cadbid.3	|- ( ph -> ( et <-> ze ) )
4	1 2	anbi12d	|- ( ph -> ( ( ps /\ th ) <-> ( ch /\ ta ) ) )
5	1 2	xorbi12d	|- ( ph -> ( ( ps \/_ th ) <-> ( ch \/_ ta ) ) )
6	3 5	anbi12d	|- ( ph -> ( ( et /\ ( ps \/_ th ) ) <-> ( ze /\ ( ch \/_ ta ) ) ) )
7	4 6	orbi12d	|- ( ph -> ( ( ( ps /\ th ) \/ ( et /\ ( ps \/_ th ) ) ) <-> ( ( ch /\ ta ) \/ ( ze /\ ( ch \/_ ta ) ) ) ) )
8		df-cad	|- ( cadd ( ps , th , et ) <-> ( ( ps /\ th ) \/ ( et /\ ( ps \/_ th ) ) ) )
9		df-cad	|- ( cadd ( ch , ta , ze ) <-> ( ( ch /\ ta ) \/ ( ze /\ ( ch \/_ ta ) ) ) )
10	7 8 9	3bitr4g	|- ( ph -> ( cadd ( ps , th , et ) <-> cadd ( ch , ta , ze ) ) )