Theorem nfneld 2614

Description: Bound-variable hypothesis builder for inequality. (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 2520	. 2 ⊢ (A ∉ B ↔ ¬ A ∈ B)
2		nfneld.1	. . . 4 ⊢ (φ → ℲxA)
3		nfneld.2	. . . 4 ⊢ (φ → ℲxB)
4	2, 3	nfeld 2505	. . 3 ⊢ (φ → Ⅎx A ∈ B)
5	4	nfnd 1791	. 2 ⊢ (φ → Ⅎx ¬ A ∈ B)
6	1, 5	nfxfrd 1571	1 ⊢ (φ → Ⅎx A ∉ B)

Syntax hints:  ¬ wn 3   → wi 4  Ⅎwnf 1544   ∈ wcel 1710  Ⅎwnfc 2477   ∉ wnel 2518

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542  df-nf 1545  df-cleq 2346  df-clel 2349  df-nfc 2479  df-nel 2520

This theorem is referenced by: (None)