ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  eluni2 GIF version

Theorem eluni2 3710
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 1572 . 2 (∃𝑥(𝐴𝑥𝑥𝐵) ↔ ∃𝑥(𝑥𝐵𝐴𝑥))
2 eluni 3709 . 2 (𝐴 𝐵 ↔ ∃𝑥(𝐴𝑥𝑥𝐵))
3 df-rex 2399 . 2 (∃𝑥𝐵 𝐴𝑥 ↔ ∃𝑥(𝑥𝐵𝐴𝑥))
41, 2, 33bitr4i 211 1 (𝐴 𝐵 ↔ ∃𝑥𝐵 𝐴𝑥)
Colors of variables: wff set class
Syntax hints:  wa 103  wb 104  wex 1453  wcel 1465  wrex 2394   cuni 3706
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-rex 2399  df-v 2662  df-uni 3707
This theorem is referenced by:  uni0b  3731  intssunim  3763  iuncom4  3790  inuni  4050  ssorduni  4373  unon  4397  cnvuni  4695  chfnrn  5499  isbasis3g  12140  eltg2b  12150  tgcl  12160  epttop  12186  txuni2  12352
  Copyright terms: Public domain W3C validator