Theorem diveq1d

Description: Equality in terms of unit ratio. (Contributed by Mario Carneiro, 27-May-2016)

	Ref	Expression
Hypotheses	div1d.1	$$⊢\phi \to A\in ℂ$$
	divcld.2	$$⊢\phi \to B\in ℂ$$
	divcld.3	$$⊢\phi \to B\ne 0$$
	diveq1d.4	$$⊢\phi \to \frac{A}{B}=1$$
Assertion	diveq1d	$$⊢\phi \to A=B$$

Proof

Step	Hyp	Ref	Expression
1		div1d.1	$$⊢\phi \to A\in ℂ$$
2		divcld.2	$$⊢\phi \to B\in ℂ$$
3		divcld.3	$$⊢\phi \to B\ne 0$$
4		diveq1d.4	$$⊢\phi \to \frac{A}{B}=1$$
5		diveq1	$$⊢\left(A\in ℂ\wedge B\in ℂ\wedge B\ne 0\right)\to \left(\frac{A}{B}=1\leftrightarrow A=B\right)$$
6	1 2 3 5	syl3anc	$$⊢\phi \to \left(\frac{A}{B}=1\leftrightarrow A=B\right)$$
7	4 6	mpbid	$$⊢\phi \to A=B$$