Theorem cadbi123i

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

	Ref	Expression
Hypotheses	cadbii.1	|- ( ph <-> ps )
	cadbii.2	|- ( ch <-> th )
	cadbii.3	|- ( ta <-> et )
Assertion	cadbi123i	|- ( cadd ( ph , ch , ta ) <-> cadd ( ps , th , et ) )

Proof

Step	Hyp	Ref	Expression
1		cadbii.1	|- ( ph <-> ps )
2		cadbii.2	|- ( ch <-> th )
3		cadbii.3	|- ( ta <-> et )
4	1	a1i	|- ( T. -> ( ph <-> ps ) )
5	2	a1i	|- ( T. -> ( ch <-> th ) )
6	3	a1i	|- ( T. -> ( ta <-> et ) )
7	4 5 6	cadbi123d	|- ( T. -> ( cadd ( ph , ch , ta ) <-> cadd ( ps , th , et ) ) )
8	7	mptru	|- ( cadd ( ph , ch , ta ) <-> cadd ( ps , th , et ) )