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Metamath Proof Explorer

Theorem subeq0ad

Description: The difference of two complex numbers is zero iff they are equal. Deduction form of subeq0 . Generalization of subeq0d . (Contributed by David Moews, 28-Feb-2017)

	Ref	Expression
Hypotheses	negidd.1	$⊢ φ \to A \in ℂ$
	pncand.2	$⊢ φ \to B \in ℂ$
Assertion	subeq0ad	$⊢ φ \to (A - B = 0 \leftrightarrow A = B)$

Proof

Step	Hyp	Ref	Expression
1		negidd.1	$⊢ φ \to A \in ℂ$
2		pncand.2	$⊢ φ \to B \in ℂ$
3		subeq0	$⊢ (A \in ℂ \land B \in ℂ) \to (A - B = 0 \leftrightarrow A = B)$
4	1 2 3	syl2anc	$⊢ φ \to (A - B = 0 \leftrightarrow A = B)$