Theorem hadbi123i

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

	Ref	Expression
Hypotheses	hadbii.1	|- ( ph <-> ps )
	hadbii.2	|- ( ch <-> th )
	hadbii.3	|- ( ta <-> et )
Assertion	hadbi123i	|- ( hadd ( ph , ch , ta ) <-> hadd ( ps , th , et ) )

Proof

Step	Hyp	Ref	Expression
1		hadbii.1	|- ( ph <-> ps )
2		hadbii.2	|- ( ch <-> th )
3		hadbii.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	hadbi123d	|- ( T. -> ( hadd ( ph , ch , ta ) <-> hadd ( ps , th , et ) ) )
8	7	mptru	|- ( hadd ( ph , ch , ta ) <-> hadd ( ps , th , et ) )