Theorem divne0b

Description: The ratio of nonzero numbers is nonzero. (Contributed by NM, 2-Aug-2004) (Proof shortened by Mario Carneiro, 27-May-2016)

	Ref	Expression
Assertion	divne0b	$$⊢\left(A\in ℂ\wedge B\in ℂ\wedge B\ne 0\right)\to \left(A\ne 0\leftrightarrow \frac{A}{B}\ne 0\right)$$

Proof

Step	Hyp	Ref	Expression
1		diveq0	$$⊢\left(A\in ℂ\wedge B\in ℂ\wedge B\ne 0\right)\to \left(\frac{A}{B}=0\leftrightarrow A=0\right)$$
2	1	bicomd	$$⊢\left(A\in ℂ\wedge B\in ℂ\wedge B\ne 0\right)\to \left(A=0\leftrightarrow \frac{A}{B}=0\right)$$
3	2	necon3bid	$$⊢\left(A\in ℂ\wedge B\in ℂ\wedge B\ne 0\right)\to \left(A\ne 0\leftrightarrow \frac{A}{B}\ne 0\right)$$