Theorem nfnel 2612

Description: Bound-variable hypothesis builder for inequality. (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 2520	. 2 ⊢ (A ∉ B ↔ ¬ A ∈ B)
2		nfnel.1	. . . 4 ⊢ ℲxA
3		nfnel.2	. . . 4 ⊢ ℲxB
4	2, 3	nfel 2498	. . 3 ⊢ Ⅎx A ∈ B
5	4	nfn 1793	. 2 ⊢ Ⅎx ¬ A ∈ B
6	1, 5	nfxfr 1570	1 ⊢ Ⅎx A ∉ B

This proof depends on syntax axioms:  ¬ wn 3  Ⅎwnf 1544   ∈ wcel 1710  Ⅎwnfc 2477   ∉ wnel 2518

This proof depends on 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-12 1925  ax-ext 2334

This proof depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-cleq 2346  df-clel 2349  df-nfc 2479  df-nel 2520

This theorem is used by: (None)