Theorem nfdif 4069

Description: Bound-variable hypothesis builder for class difference. (Contributed by NM, 3-Dec-2003.) (Revised by Mario Carneiro, 13-Oct-2016.) Avoid ax-10 2180, ax-11 2197, ax-12 2216. (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 4052	. . 3 ⊢ (𝑦 ∈ (𝐴 ∖ 𝐵) ↔ (𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ∈ 𝐵))
2		nfdif.1	. . . . 5 ⊢ Ⅎ𝑥𝐴
3	2	nfcri 2918	. . . 4 ⊢ Ⅎ𝑥 𝑦 ∈ 𝐴
4		nfdif.2	. . . . . 6 ⊢ Ⅎ𝑥𝐵
5	4	nfcri 2918	. . . . 5 ⊢ Ⅎ𝑥 𝑦 ∈ 𝐵
6	5	nfn 1890	. . . 4 ⊢ Ⅎ𝑥 ¬ 𝑦 ∈ 𝐵
7	3, 6	nfan 1934	. . 3 ⊢ Ⅎ𝑥(𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ∈ 𝐵)
8	1, 7	nfxfr 1886	. 2 ⊢ Ⅎ𝑥 𝑦 ∈ (𝐴 ∖ 𝐵)
9	8	nfci 2914	1 ⊢ Ⅎ𝑥(𝐴 ∖ 𝐵)

This proof depends on syntax axioms:  ¬ wn 3   ∧ wa 401   ∈ wcel 2147  Ⅎwnfc 2911   ∖ cdif 4048

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-ex 1813  df-nf 1817  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-v 3456  df-dif 4050

This theorem is used by:  nfsymdif  4205  csbdif  4484  iunxdif3  5060  boxcutc  9007  nfsup  9480  gsum2d2lem  20238  iunconn  23801  iundisj  25924  iundisj2  25925  limciun  26269  difrab2  33193  iundisjf  33281  iundisj2f  33282  suppss2f  33330  aciunf1  33355  iundisjfi  33484  iundisj2fi  33485  suppgsumssiun  33732  fedgmullem2  34373  sigapildsys  34906  vvdifopab  39463  compab  45685  iunconnlem2  46177  supminfxr2  46752  stoweidlem28  47311  stoweidlem34  47317  stoweidlem46  47329  stoweidlem53  47336  stoweidlem55  47338  stoweidlem59  47342  stirlinglem5  47361  preimagelt  47982  preimalegt  47983