Theorem divne0bd

Description: A ratio is zero iff the numerator is zero. (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	$$⊢\phi \to B\ne 0$$
Assertion	divne0bd	$$⊢\phi \to \left(A\ne 0\leftrightarrow \frac{A}{B}\ne 0\right)$$

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		divne0b	$$⊢\left(A\in ℂ\wedge B\in ℂ\wedge B\ne 0\right)\to \left(A\ne 0\leftrightarrow \frac{A}{B}\ne 0\right)$$
5	1 2 3 4	syl3anc	$$⊢\phi \to \left(A\ne 0\leftrightarrow \frac{A}{B}\ne 0\right)$$