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	$$⊢\phi \to A\in \mathrm{No}$$
	addsassd.2	$$⊢\phi \to B\in \mathrm{No}$$
	addsassd.3	$$⊢\phi \to C\in \mathrm{No}$$
Assertion	adds32d	$$⊢\phi \to \left(A{+}_{s}B\right){+}_{s}C=\left(A{+}_{s}C\right){+}_{s}B$$

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	2 3	addscomd	$$⊢\phi \to B{+}_{s}C=C{+}_{s}B$$
5	4	oveq2d	$$⊢\phi \to A{+}_s\left(B{+}_{s}C\right)=A{+}_s\left(C{+}_{s}B\right)$$
6	1 2 3	addsassd	$$⊢\phi \to \left(A{+}_{s}B\right){+}_{s}C=A{+}_s\left(B{+}_{s}C\right)$$
7	1 3 2	addsassd	$$⊢\phi \to \left(A{+}_{s}C\right){+}_{s}B=A{+}_s\left(C{+}_{s}B\right)$$
8	5 6 7	3eqtr4d	$$⊢\phi \to \left(A{+}_{s}B\right){+}_{s}C=\left(A{+}_{s}C\right){+}_{s}B$$