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Metamath Proof Explorer

Theorem negne0bd

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

		Ref	Expression
	Hypothesis	negidd.1	$⊢ φ \to A \in ℂ$
	Assertion	negne0bd	$⊢ φ \to (A \neq 0 \leftrightarrow - A \neq 0)$

Step	Hyp	Ref	Expression
1		negidd.1	$⊢ φ \to A \in ℂ$
2	1	negeq0d	$⊢ φ \to (A = 0 \leftrightarrow - A = 0)$
3	2	necon3bid	$⊢ φ \to (A \neq 0 \leftrightarrow - A \neq 0)$