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Metamath Proof Explorer

Theorem nfneld

Description: Bound-variable hypothesis builder for negated membership. (Contributed by David Abernethy, 26-Jun-2011) (Revised by Mario Carneiro, 7-Oct-2016)

	Ref	Expression
Hypotheses	nfneld.1	$⊢ φ \to \underline{Ⅎ} x A$
	nfneld.2	$⊢ φ \to \underline{Ⅎ} x B$
Assertion	nfneld	$⊢ φ \to Ⅎ x A \notin B$

Proof

Step	Hyp	Ref	Expression
1		nfneld.1	$⊢ φ \to \underline{Ⅎ} x A$
2		nfneld.2	$⊢ φ \to \underline{Ⅎ} x B$
3		df-nel	$⊢ A \notin B \leftrightarrow \neg A \in B$
4	1 2	nfeld	$⊢ φ \to Ⅎ x A \in B$
5	4	nfnd	$⊢ φ \to Ⅎ x \neg A \in B$
6	3 5	nfxfrd	$⊢ φ \to Ⅎ x A \notin B$