Theorem mul12i

Description: Commutative/associative law that swaps the first two factors in a triple product. (Contributed by NM, 11-May-1999) (Proof shortened by Andrew Salmon, 19-Nov-2011)

	Ref	Expression
Hypotheses	mul.1	$$⊢A\in ℂ$$
	mul.2	$$⊢B\in ℂ$$
	mul.3	$$⊢C\in ℂ$$
Assertion	mul12i	$$⊢ABC=BAC$$

Proof

Step	Hyp	Ref	Expression
1		mul.1	$$⊢A\in ℂ$$
2		mul.2	$$⊢B\in ℂ$$
3		mul.3	$$⊢C\in ℂ$$
4		mul12	$$⊢\left(A\in ℂ\wedge B\in ℂ\wedge C\in ℂ\right)\to ABC=BAC$$
5	1 2 3 4	mp3an	$$⊢ABC=BAC$$