Theorem rediveq1d

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

	Ref	Expression
Hypotheses	redivcan2d.a	⊢ ( 𝜑 → 𝐴 ∈ ℝ )
	redivcan2d.b	⊢ ( 𝜑 → 𝐵 ∈ ℝ )
	redivcan2d.z	⊢ ( 𝜑 → 𝐵 ≠ 0 )
Assertion	rediveq1d	⊢ ( 𝜑 → ( ( 𝐴 /ℝ 𝐵 ) = 1 ↔ 𝐴 = 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		redivcan2d.a	⊢ ( 𝜑 → 𝐴 ∈ ℝ )
2		redivcan2d.b	⊢ ( 𝜑 → 𝐵 ∈ ℝ )
3		redivcan2d.z	⊢ ( 𝜑 → 𝐵 ≠ 0 )
4		1red	⊢ ( 𝜑 → 1 ∈ ℝ )
5	1 4 2 3	redivmul2d	⊢ ( 𝜑 → ( ( 𝐴 /ℝ 𝐵 ) = 1 ↔ 𝐴 = ( 𝐵 · 1 ) ) )
6		ax-1rid	⊢ ( 𝐵 ∈ ℝ → ( 𝐵 · 1 ) = 𝐵 )
7	2 6	syl	⊢ ( 𝜑 → ( 𝐵 · 1 ) = 𝐵 )
8	7	eqeq2d	⊢ ( 𝜑 → ( 𝐴 = ( 𝐵 · 1 ) ↔ 𝐴 = 𝐵 ) )
9	5 8	bitrd	⊢ ( 𝜑 → ( ( 𝐴 /ℝ 𝐵 ) = 1 ↔ 𝐴 = 𝐵 ) )