Theorem nfnel 3130

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
nfnel.1	⊢ Ⅎ𝑥𝐴
nfnel.2	⊢ Ⅎ𝑥𝐵

Assertion

Ref	Expression
nfnel	⊢ Ⅎ𝑥 𝐴 ∉ 𝐵

Proof of Theorem nfnel

Step	Hyp	Ref	Expression
1		df-nel 3124	. 2 ⊢ (𝐴 ∉ 𝐵 ↔ ¬ 𝐴 ∈ 𝐵)
2		nfnel.1	. . . 4 ⊢ Ⅎ𝑥𝐴
3		nfnel.2	. . . 4 ⊢ Ⅎ𝑥𝐵
4	2, 3	nfel 2992	. . 3 ⊢ Ⅎ𝑥 𝐴 ∈ 𝐵
5	4	nfn 1853	. 2 ⊢ Ⅎ𝑥 ¬ 𝐴 ∈ 𝐵
6	1, 5	nfxfr 1849	1 ⊢ Ⅎ𝑥 𝐴 ∉ 𝐵

Syntax hints:  ¬ wn 3  Ⅎwnf 1780   ∈ wcel 2110  Ⅎwnfc 2961   ∉ wnel 3123

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2156  ax-12 2172  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1536  df-ex 1777  df-nf 1781  df-cleq 2814  df-clel 2893  df-nfc 2963  df-nel 3124

This theorem is referenced by: (None)