Theorem nfdif 4092

Description: Bound-variable hypothesis builder for class difference. (Contributed by NM, 3-Dec-2003.) (Revised by Mario Carneiro, 13-Oct-2016.) Avoid ax-10 2142, ax-11 2158, ax-12 2178. (Revised by SN, 14-May-2025.)

Hypotheses

Ref	Expression
nfdif.1	⊢ Ⅎ𝑥𝐴
nfdif.2	⊢ Ⅎ𝑥𝐵

Assertion

Ref	Expression
nfdif	⊢ Ⅎ𝑥(𝐴 ∖ 𝐵)

Proof of Theorem nfdif

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eldif 3924	. . 3 ⊢ (𝑦 ∈ (𝐴 ∖ 𝐵) ↔ (𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ∈ 𝐵))
2		nfdif.1	. . . . 5 ⊢ Ⅎ𝑥𝐴
3	2	nfcri 2883	. . . 4 ⊢ Ⅎ𝑥 𝑦 ∈ 𝐴
4		nfdif.2	. . . . . 6 ⊢ Ⅎ𝑥𝐵
5	4	nfcri 2883	. . . . 5 ⊢ Ⅎ𝑥 𝑦 ∈ 𝐵
6	5	nfn 1857	. . . 4 ⊢ Ⅎ𝑥 ¬ 𝑦 ∈ 𝐵
7	3, 6	nfan 1899	. . 3 ⊢ Ⅎ𝑥(𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ∈ 𝐵)
8	1, 7	nfxfr 1853	. 2 ⊢ Ⅎ𝑥 𝑦 ∈ (𝐴 ∖ 𝐵)
9	8	nfci 2879	1 ⊢ Ⅎ𝑥(𝐴 ∖ 𝐵)

Syntax hints:  ¬ wn 3   ∧ wa 395   ∈ wcel 2109  Ⅎwnfc 2876   ∖ cdif 3911

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1543  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-v 3449  df-dif 3917

This theorem is referenced by:  nfsymdif  4220  csbdif  4487  iunxdif3  5059  boxcutc  8914  nfsup  9402  gsum2d2lem  19903  iunconn  23315  iundisj  25449  iundisj2  25450  limciun  25795  difrab2  32427  iundisjf  32518  iundisj2f  32519  suppss2f  32562  aciunf1  32587  iundisjfi  32719  iundisj2fi  32720  fedgmullem2  33626  sigapildsys  34152  vvdifopab  38249  compab  44431  iunconnlem2  44924  supminfxr2  45465  stoweidlem28  46026  stoweidlem34  46032  stoweidlem46  46044  stoweidlem53  46051  stoweidlem55  46053  stoweidlem59  46057  stirlinglem5  46076  preimagelt  46697  preimalegt  46698