Theorem redivcan3d

Description: A cancellation law for division. (Contributed by SN, 25-Nov-2025)

	Ref	Expression
Hypotheses	redivcan2d.a	$$⊢\phi \to A\in ℝ$$
	redivcan2d.b	$$⊢\phi \to B\in ℝ$$
	redivcan2d.z	$$⊢\phi \to B\ne 0$$
Assertion	redivcan3d	$$⊢\phi \to BA{/}_{ℝ}B=A$$

Proof

Step	Hyp	Ref	Expression
1		redivcan2d.a	$$⊢\phi \to A\in ℝ$$
2		redivcan2d.b	$$⊢\phi \to B\in ℝ$$
3		redivcan2d.z	$$⊢\phi \to B\ne 0$$
4		eqidd	$$⊢\phi \to BA=BA$$
5	2 1	remulcld	$$⊢\phi \to BA\in ℝ$$
6	5 1 2 3	redivmuld	$$⊢\phi \to \left(BA{/}_{ℝ}B=A\leftrightarrow BA=BA\right)$$
7	4 6	mpbird	$$⊢\phi \to BA{/}_{ℝ}B=A$$