Theorem nfneld 2439

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 2432	. 2 ⊢ (𝐴 ∉ 𝐵 ↔ ¬ 𝐴 ∈ 𝐵)
2		nfneld.1	. . . 4 ⊢ (𝜑 → Ⅎ𝑥𝐴)
3		nfneld.2	. . . 4 ⊢ (𝜑 → Ⅎ𝑥𝐵)
4	2, 3	nfeld 2324	. . 3 ⊢ (𝜑 → Ⅎ𝑥 𝐴 ∈ 𝐵)
5	4	nfnd 1645	. 2 ⊢ (𝜑 → Ⅎ𝑥 ¬ 𝐴 ∈ 𝐵)
6	1, 5	nfxfrd 1463	1 ⊢ (𝜑 → Ⅎ𝑥 𝐴 ∉ 𝐵)

Syntax hints:  ¬ wn 3   → wi 4  Ⅎwnf 1448   ∈ wcel 2136  Ⅎwnfc 2295   ∉ wnel 2431

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-4 1498  ax-17 1514  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-fal 1349  df-nf 1449  df-cleq 2158  df-clel 2161  df-nfc 2297  df-nel 2432

This theorem is referenced by: (None)