Theorem nfdif 4081

Description: Bound-variable hypothesis builder for class difference. (Contributed by NM, 3-Dec-2003.) (Revised by Mario Carneiro, 13-Oct-2016.) Avoid ax-10 2146, ax-11 2162, ax-12 2184. (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 3911	. . 3 ⊢ (𝑦 ∈ (𝐴 ∖ 𝐵) ↔ (𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ∈ 𝐵))
2		nfdif.1	. . . . 5 ⊢ Ⅎ𝑥𝐴
3	2	nfcri 2890	. . . 4 ⊢ Ⅎ𝑥 𝑦 ∈ 𝐴
4		nfdif.2	. . . . . 6 ⊢ Ⅎ𝑥𝐵
5	4	nfcri 2890	. . . . 5 ⊢ Ⅎ𝑥 𝑦 ∈ 𝐵
6	5	nfn 1858	. . . 4 ⊢ Ⅎ𝑥 ¬ 𝑦 ∈ 𝐵
7	3, 6	nfan 1900	. . 3 ⊢ Ⅎ𝑥(𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ∈ 𝐵)
8	1, 7	nfxfr 1854	. 2 ⊢ Ⅎ𝑥 𝑦 ∈ (𝐴 ∖ 𝐵)
9	8	nfci 2886	1 ⊢ Ⅎ𝑥(𝐴 ∖ 𝐵)

Syntax hints:  ¬ wn 3   ∧ wa 395   ∈ wcel 2113  Ⅎwnfc 2883   ∖ cdif 3898

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1544  df-ex 1781  df-nf 1785  df-sb 2068  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-v 3442  df-dif 3904

This theorem is referenced by:  nfsymdif  4209  csbdif  4478  iunxdif3  5050  boxcutc  8879  nfsup  9354  gsum2d2lem  19902  iunconn  23372  iundisj  25505  iundisj2  25506  limciun  25851  difrab2  32572  iundisjf  32664  iundisj2f  32665  suppss2f  32716  aciunf1  32741  iundisjfi  32876  iundisj2fi  32877  fedgmullem2  33787  sigapildsys  34319  vvdifopab  38454  compab  44678  iunconnlem2  45171  supminfxr2  45709  stoweidlem28  46268  stoweidlem34  46274  stoweidlem46  46286  stoweidlem53  46293  stoweidlem55  46295  stoweidlem59  46299  stirlinglem5  46318  preimagelt  46939  preimalegt  46940