Theorem int-mulcomd

Description: MultiplicationCommutativity generator rule. (Contributed by Stanislas Polu, 7-Apr-2020)

	Ref	Expression
Hypotheses	int-mulcomd.1	$$⊢\phi \to B\in ℝ$$
	int-mulcomd.2	$$⊢\phi \to C\in ℝ$$
	int-mulcomd.3	$$⊢\phi \to A=B$$
Assertion	int-mulcomd	$$⊢\phi \to BC=CA$$

Proof

Step	Hyp	Ref	Expression
1		int-mulcomd.1	$$⊢\phi \to B\in ℝ$$
2		int-mulcomd.2	$$⊢\phi \to C\in ℝ$$
3		int-mulcomd.3	$$⊢\phi \to A=B$$
4	1	recnd	$$⊢\phi \to B\in ℂ$$
5	2	recnd	$$⊢\phi \to C\in ℂ$$
6	4 5	mulcomd	$$⊢\phi \to BC=CB$$
7	3	eqcomd	$$⊢\phi \to B=A$$
8	7	oveq2d	$$⊢\phi \to CB=CA$$
9	6 8	eqtrd	$$⊢\phi \to BC=CA$$