Theorem had1

Description: If the first input is true, then the adder sum is equivalent to the biconditionality of the other two inputs, and conversely. (Contributed by Mario Carneiro, 4-Sep-2016) (Proof shortened by Wolf Lammen, 11-Jul-2020) Strengthen to a biconditional. (Revised by BJ, 10-Aug-2026)

	Ref	Expression
Assertion	had1	|- ( ph <-> ( hadd ( ph , ps , ch ) <-> ( ps <-> ch ) ) )

Proof

Step	Hyp	Ref	Expression
1		hadrot	|- ( hadd ( ph , ps , ch ) <-> hadd ( ps , ch , ph ) )
2		hadbi	|- ( hadd ( ps , ch , ph ) <-> ( ( ps <-> ch ) <-> ph ) )
3	1 2	bitri	|- ( hadd ( ph , ps , ch ) <-> ( ( ps <-> ch ) <-> ph ) )
4		birot	|- ( ( ph <-> ( hadd ( ph , ps , ch ) <-> ( ps <-> ch ) ) ) <-> ( hadd ( ph , ps , ch ) <-> ( ( ps <-> ch ) <-> ph ) ) )
5	3 4	mpbir	|- ( ph <-> ( hadd ( ph , ps , ch ) <-> ( ps <-> ch ) ) )