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	$$⊢\phi \to A\in \mathrm{No}$$
	addsassd.2	$$⊢\phi \to B\in \mathrm{No}$$
	addsassd.3	$$⊢\phi \to C\in \mathrm{No}$$
Assertion	adds12d	Could not format assertion : No typesetting found for |- ( ph -> ( A +s ( B +s C ) ) = ( B +s ( A +s C ) ) ) with typecode |-

Proof

Step	Hyp	Ref	Expression
1		addsassd.1	$$⊢\phi \to A\in \mathrm{No}$$
2		addsassd.2	$$⊢\phi \to B\in \mathrm{No}$$
3		addsassd.3	$$⊢\phi \to C\in \mathrm{No}$$
4	1 2	addscomd	Could not format ( ph -> ( A +s B ) = ( B +s A ) ) : No typesetting found for |- ( ph -> ( A +s B ) = ( B +s A ) ) with typecode |-
5	4	oveq1d	Could not format ( ph -> ( ( A +s B ) +s C ) = ( ( B +s A ) +s C ) ) : No typesetting found for |- ( ph -> ( ( A +s B ) +s C ) = ( ( B +s A ) +s C ) ) with typecode |-
6	1 2 3	addsassd	Could not format ( ph -> ( ( A +s B ) +s C ) = ( A +s ( B +s C ) ) ) : No typesetting found for |- ( ph -> ( ( A +s B ) +s C ) = ( A +s ( B +s C ) ) ) with typecode |-
7	2 1 3	addsassd	Could not format ( ph -> ( ( B +s A ) +s C ) = ( B +s ( A +s C ) ) ) : No typesetting found for |- ( ph -> ( ( B +s A ) +s C ) = ( B +s ( A +s C ) ) ) with typecode |-
8	5 6 7	3eqtr3d	Could not format ( ph -> ( A +s ( B +s C ) ) = ( B +s ( A +s C ) ) ) : No typesetting found for |- ( ph -> ( A +s ( B +s C ) ) = ( B +s ( A +s C ) ) ) with typecode |-