Theorem nfneld 3057

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

Hypotheses

Ref	Expression
nfneld.1	⊢ (𝜑 → Ⅎ𝑥𝐴)
nfneld.2	⊢ (𝜑 → Ⅎ𝑥𝐵)

Assertion

Ref	Expression
nfneld	⊢ (𝜑 → Ⅎ𝑥 𝐴 ∉ 𝐵)

Proof of Theorem nfneld

Step	Hyp	Ref	Expression
1		df-nel 3050	. 2 ⊢ (𝐴 ∉ 𝐵 ↔ ¬ 𝐴 ∈ 𝐵)
2		nfneld.1	. . . 4 ⊢ (𝜑 → Ⅎ𝑥𝐴)
3		nfneld.2	. . . 4 ⊢ (𝜑 → Ⅎ𝑥𝐵)
4	2, 3	nfeld 2918	. . 3 ⊢ (𝜑 → Ⅎ𝑥 𝐴 ∈ 𝐵)
5	4	nfnd 1861	. 2 ⊢ (𝜑 → Ⅎ𝑥 ¬ 𝐴 ∈ 𝐵)
6	1, 5	nfxfrd 1856	1 ⊢ (𝜑 → Ⅎ𝑥 𝐴 ∉ 𝐵)

Syntax hints:  ¬ wn 3   → wi 4  Ⅎwnf 1786   ∈ wcel 2106  Ⅎwnfc 2887   ∉ wnel 3049

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-ex 1783  df-nf 1787  df-cleq 2730  df-clel 2816  df-nfc 2889  df-nel 3050

This theorem is referenced by: (None)