Theorem rabid2 3428

Description: An "identity" law for restricted class abstraction. Prefer rabid2im 3427 if one direction is sufficient. (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 2894	. 2 ⊢ Ⅎ𝑥𝐴
2	1	rabid2f 3426	1 ⊢ (𝐴 = {𝑥 ∈ 𝐴 ∣ 𝜑} ↔ ∀𝑥 ∈ 𝐴 𝜑)

Syntax hints:   ↔ wb 206   = wceq 1541  ∀wral 3047  {crab 3395

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 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703

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 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ral 3048  df-rab 3396

This theorem is referenced by:  iinrab2  5016  riinrab  5030  dmmptg  6189  frpoinsg  6290  dmmptd  6626  fneqeql  6979  fmpt  7043  tfisg  7784  zfrep6  7887  frinsg  9644  axdc2lem  10339  ioomax  13322  iccmax  13323  hashbc  14360  lcmf0  16545  dfphi2  16685  phiprmpw  16687  phisum  16702  isnsg4  19079  symggen2  19383  psgnfvalfi  19425  lssuni  20872  psgnghm2  21518  ocv0  21614  dsmmfi  21675  frlmfibas  21699  frlmlbs  21734  psr1baslem  22097  ordtrest2lem  23118  comppfsc  23447  xkouni  23514  xkoccn  23534  tsmsfbas  24043  clsocv  25177  ehlbase  25342  ovolicc2lem4  25448  itg2monolem1  25678  musum  27128  lgsquadlem2  27319  umgr2v2evd2  29506  frgrregorufr0  30304  ubthlem1  30850  xrsclat  32992  psgndmfi  33067  primefldgen1  33287  zarcls0  33881  ordtrest2NEWlem  33935  hasheuni  34098  measvuni  34227  imambfm  34275  subfacp1lem6  35229  connpconn  35279  cvmliftmolem2  35326  cvmlift2lem12  35358  poimirlem28  37698  fdc  37795  isbnd3  37834  pmap1N  39876  pol1N  40019  dia1N  41162  dihwN  41398  vdioph  42882  fiphp3d  42922  stirlinglem14  46195  fvmptrabdm  47403  suppdm  48621