Theorem adds12d

Description: Commutative/associative law that swaps the first two terms in a triple sum. (Contributed by Scott Fenton, 9-Mar-2025)

	Ref	Expression
Hypotheses	addsassd.1	|- ( ph -> A e. No )
	addsassd.2	|- ( ph -> B e. No )
	addsassd.3	|- ( ph -> C e. No )
Assertion	adds12d	|- ( ph -> ( A +s ( B +s C ) ) = ( B +s ( A +s C ) ) )

Proof

Step	Hyp	Ref	Expression
1		addsassd.1	|- ( ph -> A e. No )
2		addsassd.2	|- ( ph -> B e. No )
3		addsassd.3	|- ( ph -> C e. No )
4	1 2	addscomd	|- ( ph -> ( A +s B ) = ( B +s A ) )
5	4	oveq1d	|- ( ph -> ( ( A +s B ) +s C ) = ( ( B +s A ) +s C ) )
6	1 2 3	addsassd	|- ( ph -> ( ( A +s B ) +s C ) = ( A +s ( B +s C ) ) )
7	2 1 3	addsassd	|- ( ph -> ( ( B +s A ) +s C ) = ( B +s ( A +s C ) ) )
8	5 6 7	3eqtr3d	|- ( ph -> ( A +s ( B +s C ) ) = ( B +s ( A +s C ) ) )