Theorem hadcoma

Description: Commutative law for the adder sum. (Contributed by Mario Carneiro, 4-Sep-2016) (Proof shortened by Wolf Lammen, 17-Dec-2023)

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

Proof

Step	Hyp	Ref	Expression
1		bicom	|- ( ( ph <-> ps ) <-> ( ps <-> ph ) )
2	1	bibi1i	|- ( ( ( ph <-> ps ) <-> ch ) <-> ( ( ps <-> ph ) <-> ch ) )
3		hadbi	|- ( hadd ( ph , ps , ch ) <-> ( ( ph <-> ps ) <-> ch ) )
4		hadbi	|- ( hadd ( ps , ph , ch ) <-> ( ( ps <-> ph ) <-> ch ) )
5	2 3 4	3bitr4i	|- ( hadd ( ph , ps , ch ) <-> hadd ( ps , ph , ch ) )