Theorem rabid2 3333

Description: An "identity" law for restricted class abstraction. (Contributed by NM, 9-Oct-2003.) (Proof shortened by Andrew Salmon, 30-May-2011.) (Proof shortened by Wolf Lammen, 24-Nov-2024.)

Assertion

Ref	Expression
rabid2	⊢ (𝐴 = {𝑥 ∈ 𝐴 ∣ 𝜑} ↔ ∀𝑥 ∈ 𝐴 𝜑)

Distinct variable group:   𝑥,𝐴

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem rabid2

Step	Hyp	Ref	Expression
1		nfcv 2905	. 2 ⊢ Ⅎ𝑥𝐴
2	1	rabid2f 3332	1 ⊢ (𝐴 = {𝑥 ∈ 𝐴 ∣ 𝜑} ↔ ∀𝑥 ∈ 𝐴 𝜑)

Syntax hints:   ↔ wb 205   = wceq 1539  ∀wral 3062  {crab 3303

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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2707

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-tru 1542  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2887  df-ral 3063  df-rab 3306

This theorem is referenced by:  rabxm  4326  iinrab2  5006  riinrab  5020  class2seteq  5286  dmmptg  6160  frpoinsg  6261  wfisgOLD  6272  dmmptd  6608  fneqeql  6955  fmpt  7016  zfrep6  7829  frinsg  9557  axdc2lem  10254  ioomax  13204  iccmax  13205  hashbc  14214  lcmf0  16388  dfphi2  16524  phiprmpw  16526  phisum  16540  isnsg4  18844  symggen2  19128  psgnfvalfi  19170  lssuni  20250  psgnghm2  20835  ocv0  20931  dsmmfi  20994  frlmfibas  21018  frlmlbs  21053  psr1baslem  21405  ordtrest2lem  22403  comppfsc  22732  xkouni  22799  xkoccn  22819  tsmsfbas  23328  clsocv  24463  ehlbase  24628  ovolicc2lem4  24733  itg2monolem1  24964  musum  26389  lgsquadlem2  26578  umgr2v2evd2  27943  frgrregorufr0  28737  ubthlem1  29281  xrsclat  31338  psgndmfi  31414  zarcls0  31867  ordtrest2NEWlem  31921  hasheuni  32102  measvuni  32231  imambfm  32278  subfacp1lem6  33196  connpconn  33246  cvmliftmolem2  33293  cvmlift2lem12  33325  tfisg  33835  poimirlem28  35853  fdc  35951  isbnd3  35990  pmap1N  37981  pol1N  38124  dia1N  39267  dihwN  39503  vdioph  40796  fiphp3d  40836  stirlinglem14  43857  fvmptrabdm  45029  suppdm  46095