Theorem nfdif 4152

Description: Bound-variable hypothesis builder for class difference. (Contributed by NM, 3-Dec-2003.) (Revised by Mario Carneiro, 13-Oct-2016.) Avoid ax-10 2141, 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 3986	. . 3 ⊢ (𝑦 ∈ (𝐴 ∖ 𝐵) ↔ (𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ∈ 𝐵))
2		nfdif.1	. . . . 5 ⊢ Ⅎ𝑥𝐴
3	2	nfcri 2900	. . . 4 ⊢ Ⅎ𝑥 𝑦 ∈ 𝐴
4		nfdif.2	. . . . . 6 ⊢ Ⅎ𝑥𝐵
5	4	nfcri 2900	. . . . 5 ⊢ Ⅎ𝑥 𝑦 ∈ 𝐵
6	5	nfn 1856	. . . 4 ⊢ Ⅎ𝑥 ¬ 𝑦 ∈ 𝐵
7	3, 6	nfan 1898	. . 3 ⊢ Ⅎ𝑥(𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ∈ 𝐵)
8	1, 7	nfxfr 1851	. 2 ⊢ Ⅎ𝑥 𝑦 ∈ (𝐴 ∖ 𝐵)
9	8	nfci 2896	1 ⊢ Ⅎ𝑥(𝐴 ∖ 𝐵)

Syntax hints:  ¬ wn 3   ∧ wa 395   ∈ wcel 2108  Ⅎwnfc 2893   ∖ cdif 3973

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-tru 1540  df-ex 1778  df-nf 1782  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-v 3490  df-dif 3979

This theorem is referenced by:  nfsymdif  4276  csbdif  4547  iunxdif3  5118  boxcutc  8999  nfsup  9520  gsum2d2lem  20015  iunconn  23457  iundisj  25602  iundisj2  25603  limciun  25949  difrab2  32526  iundisjf  32611  iundisj2f  32612  suppss2f  32657  aciunf1  32681  iundisjfi  32801  iundisj2fi  32802  fedgmullem2  33643  sigapildsys  34126  vvdifopab  38216  compab  44411  iunconnlem2  44906  supminfxr2  45384  stoweidlem28  45949  stoweidlem34  45955  stoweidlem46  45967  stoweidlem53  45974  stoweidlem55  45976  stoweidlem59  45980  stirlinglem5  45999  preimagelt  46620  preimalegt  46621