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Metamath Proof Explorer

Theorem subne0ad

Description: If the difference of two complex numbers is nonzero, they are unequal. Converse of subne0d . Contrapositive of subeq0bd . (Contributed by David Moews, 28-Feb-2017)

	Ref	Expression
Hypotheses	negidd.1	$⊢ φ \to A \in ℂ$
	pncand.2	$⊢ φ \to B \in ℂ$
	subne0ad.3	$⊢ φ \to A - B \neq 0$
Assertion	subne0ad	$⊢ φ \to A \neq B$

Proof

Step	Hyp	Ref	Expression
1		negidd.1	$⊢ φ \to A \in ℂ$
2		pncand.2	$⊢ φ \to B \in ℂ$
3		subne0ad.3	$⊢ φ \to A - B \neq 0$
4	1 2	subeq0ad	$⊢ φ \to (A - B = 0 \leftrightarrow A = B)$
5	4	necon3bid	$⊢ φ \to (A - B \neq 0 \leftrightarrow A \neq B)$
6	3 5	mpbid	$⊢ φ \to A \neq B$