Theorem nfdif 4070

Description: Bound-variable hypothesis builder for class difference. (Contributed by NM, 3-Dec-2003.) (Revised by Mario Carneiro, 13-Oct-2016.) Avoid ax-10 2147, ax-11 2163, ax-12 2185. (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 3900	. . 3 ⊢ (𝑦 ∈ (𝐴 ∖ 𝐵) ↔ (𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ∈ 𝐵))
2		nfdif.1	. . . . 5 ⊢ Ⅎ𝑥𝐴
3	2	nfcri 2891	. . . 4 ⊢ Ⅎ𝑥 𝑦 ∈ 𝐴
4		nfdif.2	. . . . . 6 ⊢ Ⅎ𝑥𝐵
5	4	nfcri 2891	. . . . 5 ⊢ Ⅎ𝑥 𝑦 ∈ 𝐵
6	5	nfn 1859	. . . 4 ⊢ Ⅎ𝑥 ¬ 𝑦 ∈ 𝐵
7	3, 6	nfan 1901	. . 3 ⊢ Ⅎ𝑥(𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ∈ 𝐵)
8	1, 7	nfxfr 1855	. 2 ⊢ Ⅎ𝑥 𝑦 ∈ (𝐴 ∖ 𝐵)
9	8	nfci 2887	1 ⊢ Ⅎ𝑥(𝐴 ∖ 𝐵)

Syntax hints:  ¬ wn 3   ∧ wa 395   ∈ wcel 2114  Ⅎwnfc 2884   ∖ cdif 3887

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1545  df-ex 1782  df-nf 1786  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-v 3432  df-dif 3893

This theorem is referenced by:  nfsymdif  4198  csbdif  4466  iunxdif3  5038  boxcutc  8882  nfsup  9357  gsum2d2lem  19939  iunconn  23403  iundisj  25525  iundisj2  25526  limciun  25871  difrab2  32582  iundisjf  32674  iundisj2f  32675  suppss2f  32726  aciunf1  32751  iundisjfi  32884  iundisj2fi  32885  suppgsumssiun  33148  fedgmullem2  33790  sigapildsys  34322  vvdifopab  38600  compab  44886  iunconnlem2  45379  supminfxr2  45915  stoweidlem28  46474  stoweidlem34  46480  stoweidlem46  46492  stoweidlem53  46499  stoweidlem55  46501  stoweidlem59  46505  stirlinglem5  46524  preimagelt  47145  preimalegt  47146