Theorem cjne0d

Description: A number is nonzero iff its complex conjugate is nonzero. (Contributed by Mario Carneiro, 29-May-2016)

	Ref	Expression
Hypotheses	recld.1	$$⊢\phi \to A\in ℂ$$
	cjne0d.2	$$⊢\phi \to A\ne 0$$
Assertion	cjne0d	$$⊢\phi \to \bar{A}\ne 0$$

Proof

Step	Hyp	Ref	Expression
1		recld.1	$$⊢\phi \to A\in ℂ$$
2		cjne0d.2	$$⊢\phi \to A\ne 0$$
3		cjne0	$$⊢A\in ℂ\to \left(A\ne 0\leftrightarrow \bar{A}\ne 0\right)$$
4	1 3	syl	$$⊢\phi \to \left(A\ne 0\leftrightarrow \bar{A}\ne 0\right)$$
5	2 4	mpbid	$$⊢\phi \to \bar{A}\ne 0$$