Theorem nfneld 2430

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 2423	. 2 ⊢ (𝐴 ∉ 𝐵 ↔ ¬ 𝐴 ∈ 𝐵)
2		nfneld.1	. . . 4 ⊢ (𝜑 → Ⅎ𝑥𝐴)
3		nfneld.2	. . . 4 ⊢ (𝜑 → Ⅎ𝑥𝐵)
4	2, 3	nfeld 2315	. . 3 ⊢ (𝜑 → Ⅎ𝑥 𝐴 ∈ 𝐵)
5	4	nfnd 1637	. 2 ⊢ (𝜑 → Ⅎ𝑥 ¬ 𝐴 ∈ 𝐵)
6	1, 5	nfxfrd 1455	1 ⊢ (𝜑 → Ⅎ𝑥 𝐴 ∉ 𝐵)

Syntax hints:  ¬ wn 3   → wi 4  Ⅎwnf 1440   ∈ wcel 2128  Ⅎwnfc 2286   ∉ wnel 2422

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-4 1490  ax-17 1506  ax-ial 1514  ax-i5r 1515  ax-ext 2139

This theorem depends on definitions:  df-bi 116  df-tru 1338  df-fal 1341  df-nf 1441  df-cleq 2150  df-clel 2153  df-nfc 2288  df-nel 2423

This theorem is referenced by: (None)