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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 → Ⅎ 𝑥 𝐴 )
	nfneld.2	⊢ ( 𝜑 → Ⅎ 𝑥 𝐵 )
Assertion	nfneld	⊢ ( 𝜑 → Ⅎ 𝑥 𝐴 ∉ 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		nfneld.1	⊢ ( 𝜑 → Ⅎ 𝑥 𝐴 )
2		nfneld.2	⊢ ( 𝜑 → Ⅎ 𝑥 𝐵 )
3		df-nel	⊢ ( 𝐴 ∉ 𝐵 ↔ ¬ 𝐴 ∈ 𝐵 )
4	1 2	nfeld	⊢ ( 𝜑 → Ⅎ 𝑥 𝐴 ∈ 𝐵 )
5	4	nfnd	⊢ ( 𝜑 → Ⅎ 𝑥 ¬ 𝐴 ∈ 𝐵 )
6	3 5	nfxfrd	⊢ ( 𝜑 → Ⅎ 𝑥 𝐴 ∉ 𝐵 )