Theorem nfneld 3049

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 3041	. 2 ⊢ (𝐴 ∉ 𝐵 ↔ ¬ 𝐴 ∈ 𝐵)
2		nfneld.1	. . . 4 ⊢ (𝜑 → Ⅎ𝑥𝐴)
3		nfneld.2	. . . 4 ⊢ (𝜑 → Ⅎ𝑥𝐵)
4	2, 3	nfeld 2914	. . 3 ⊢ (𝜑 → Ⅎ𝑥 𝐴 ∈ 𝐵)
5	4	nfnd 1866	. 2 ⊢ (𝜑 → Ⅎ𝑥 ¬ 𝐴 ∈ 𝐵)
6	1, 5	nfxfrd 1862	1 ⊢ (𝜑 → Ⅎ𝑥 𝐴 ∉ 𝐵)

Syntax hints:  ¬ wn 3   → wi 4  Ⅎwnf 1791   ∈ wcel 2121  Ⅎwnfc 2888   ∉ wnel 3040

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-10 2154  ax-11 2170  ax-12 2191  ax-ext 2713

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-ex 1788  df-nf 1792  df-cleq 2733  df-clel 2816  df-nfc 2890  df-nel 3041

This theorem is referenced by: (None)