Theorem add12d

Description: Commutative/associative law that swaps the first two terms in a triple sum. (Contributed by Mario Carneiro, 27-May-2016)

	Ref	Expression
Hypotheses	addd.1	$$⊢\phi \to A\in ℂ$$
	addd.2	$$⊢\phi \to B\in ℂ$$
	addd.3	$$⊢\phi \to C\in ℂ$$
Assertion	add12d	$$⊢\phi \to A+B+C=B+A+C$$

Proof

Step	Hyp	Ref	Expression
1		addd.1	$$⊢\phi \to A\in ℂ$$
2		addd.2	$$⊢\phi \to B\in ℂ$$
3		addd.3	$$⊢\phi \to C\in ℂ$$
4		add12	$$⊢\left(A\in ℂ\wedge B\in ℂ\wedge C\in ℂ\right)\to A+B+C=B+A+C$$
5	1 2 3 4	syl3anc	$$⊢\phi \to A+B+C=B+A+C$$