Theorem negne0bd

Description: A number is nonzero iff its negative is nonzero. (Contributed by Mario Carneiro, 27-May-2016)

	Ref	Expression
Hypothesis	negidd.1	⊢ ( 𝜑 → 𝐴 ∈ ℂ )
Assertion	negne0bd	⊢ ( 𝜑 → ( 𝐴 ≠ 0 ↔ - 𝐴 ≠ 0 ) )

Proof

Step	Hyp	Ref	Expression
1		negidd.1	⊢ ( 𝜑 → 𝐴 ∈ ℂ )
2	1	negeq0d	⊢ ( 𝜑 → ( 𝐴 = 0 ↔ - 𝐴 = 0 ) )
3	2	necon3bid	⊢ ( 𝜑 → ( 𝐴 ≠ 0 ↔ - 𝐴 ≠ 0 ) )