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	$$⊢\phi \to A{+}_s\left(B{+}_{s}C\right)=B{+}_s\left(A{+}_{s}C\right)$$

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	$$⊢\phi \to A{+}_{s}B=B{+}_{s}A$$
5	4	oveq1d	$$⊢\phi \to \left(A{+}_{s}B\right){+}_{s}C=\left(B{+}_{s}A\right){+}_{s}C$$
6	1 2 3	addsassd	$$⊢\phi \to \left(A{+}_{s}B\right){+}_{s}C=A{+}_s\left(B{+}_{s}C\right)$$
7	2 1 3	addsassd	$$⊢\phi \to \left(B{+}_{s}A\right){+}_{s}C=B{+}_s\left(A{+}_{s}C\right)$$
8	5 6 7	3eqtr3d	$$⊢\phi \to A{+}_s\left(B{+}_{s}C\right)=B{+}_s\left(A{+}_{s}C\right)$$