Theorem adds32d

Description: Commutative/associative law that swaps the last two terms in a triple sum. (Contributed by Scott Fenton, 22-Jan-2025)

	Ref	Expression
Hypotheses	addsassd.1	|- ( ph -> A e. No )
	addsassd.2	|- ( ph -> B e. No )
	addsassd.3	|- ( ph -> C e. No )
Assertion	adds32d	|- ( ph -> ( ( A +s B ) +s C ) = ( ( A +s C ) +s B ) )

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	2 3	addscomd	|- ( ph -> ( B +s C ) = ( C +s B ) )
5	4	oveq2d	|- ( ph -> ( A +s ( B +s C ) ) = ( A +s ( C +s B ) ) )
6	1 2 3	addsassd	|- ( ph -> ( ( A +s B ) +s C ) = ( A +s ( B +s C ) ) )
7	1 3 2	addsassd	|- ( ph -> ( ( A +s C ) +s B ) = ( A +s ( C +s B ) ) )
8	5 6 7	3eqtr4d	|- ( ph -> ( ( A +s B ) +s C ) = ( ( A +s C ) +s B ) )