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Theorem eluni2 4834
Description: Membership in class union. Restricted quantifier version. (Contributed by NM, 31-Aug-1999.)
Assertion
Ref Expression
eluni2 (𝐴 𝐵 ↔ ∃𝑥𝐵 𝐴𝑥)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem eluni2
StepHypRef Expression
1 exancom 1852 . 2 (∃𝑥(𝐴𝑥𝑥𝐵) ↔ ∃𝑥(𝑥𝐵𝐴𝑥))
2 eluni 4833 . 2 (𝐴 𝐵 ↔ ∃𝑥(𝐴𝑥𝑥𝐵))
3 df-rex 3141 . 2 (∃𝑥𝐵 𝐴𝑥 ↔ ∃𝑥(𝑥𝐵𝐴𝑥))
41, 2, 33bitr4i 304 1 (𝐴 𝐵 ↔ ∃𝑥𝐵 𝐴𝑥)
Colors of variables: wff setvar class
Syntax hints:  wb 207  wa 396  wex 1771  wcel 2105  wrex 3136   cuni 4830
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-rex 3141  df-v 3494  df-uni 4831
This theorem is referenced by:  uni0b  4855  intssuni  4889  iuncom4  4918  inuni  5237  cnvuni  5750  chfnrn  6811  ssorduni  7489  unon  7535  limuni3  7556  onfununi  7967  oarec  8177  uniinqs  8366  fissuni  8817  finsschain  8819  r1sdom  9191  rankuni2b  9270  cflm  9660  coflim  9671  axdc3lem2  9861  fpwwe2lem12  10051  uniwun  10150  tskr1om2  10178  tskuni  10193  axgroth3  10241  inaprc  10246  tskmval  10249  tskmcl  10251  suplem1pr  10462  lbsextlem2  19860  lbsextlem3  19861  isbasis3g  21485  eltg2b  21495  tgcl  21505  ppttop  21543  epttop  21545  neiptoptop  21667  tgcmp  21937  locfincmp  22062  dissnref  22064  comppfsc  22068  1stckgenlem  22089  txuni2  22101  txcmplem1  22177  tgqtop  22248  filuni  22421  alexsubALTlem4  22586  ptcmplem3  22590  ptcmplem4  22591  utoptop  22770  icccmplem1  23357  icccmplem3  23359  cnheibor  23486  bndth  23489  lebnumlem1  23492  bcthlem4  23857  ovolicc2lem5  24049  dyadmbllem  24127  itg2gt0  24288  rexunirn  30183  unipreima  30319  acunirnmpt2  30333  acunirnmpt2f  30334  reff  31002  locfinreflem  31003  cmpcref  31013  ddemeas  31394  dya2iocuni  31440  bnj1379  32001  cvmsss2  32418  cvmseu  32420  untuni  32832  dfon2lem3  32927  dfon2lem7  32931  dfon2lem8  32932  frrlem9  33028  brbigcup  33256  neibastop1  33604  neibastop2lem  33605  fvineqsneq  34575  heicant  34808  mblfinlem1  34810  cover2  34870  heiborlem9  34978  unichnidl  35190  erim2  35791  prtlem16  35885  prter2  35897  prter3  35898  restuni3  41261  disjinfi  41330  cncfuni  42045  intsaluni  42489  salgencntex  42503
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