Theorem rediveq1d

Description: Equality in terms of unit ratio. (Contributed by SN, 2-Apr-2026)

	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	rediveq1d	$$⊢\phi \to \left(A{/}_{ℝ}B=1\leftrightarrow A=B\right)$$

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		1red	$$⊢\phi \to 1\in ℝ$$
5	1 4 2 3	redivmul2d	$$⊢\phi \to \left(A{/}_{ℝ}B=1\leftrightarrow A=B\cdot 1\right)$$
6		ax-1rid	$$⊢B\in ℝ\to B\cdot 1=B$$
7	2 6	syl	$$⊢\phi \to B\cdot 1=B$$
8	7	eqeq2d	$$⊢\phi \to \left(A=B\cdot 1\leftrightarrow A=B\right)$$
9	5 8	bitrd	$$⊢\phi \to \left(A{/}_{ℝ}B=1\leftrightarrow A=B\right)$$