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

Theorem eluni2 3828
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 1619 . 2 (∃𝑥(𝐴𝑥𝑥𝐵) ↔ ∃𝑥(𝑥𝐵𝐴𝑥))
2 eluni 3827 . 2 (𝐴 𝐵 ↔ ∃𝑥(𝐴𝑥𝑥𝐵))
3 df-rex 2474 . 2 (∃𝑥𝐵 𝐴𝑥 ↔ ∃𝑥(𝑥𝐵𝐴𝑥))
41, 2, 33bitr4i 212 1 (𝐴 𝐵 ↔ ∃𝑥𝐵 𝐴𝑥)
Colors of variables: wff set class
Syntax hints:  wa 104  wb 105  wex 1503  wcel 2160  wrex 2469   cuni 3824
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-rex 2474  df-v 2754  df-uni 3825
This theorem is referenced by:  uni0b  3849  intssunim  3881  iuncom4  3908  inuni  4173  ssorduni  4504  unon  4528  cnvuni  4831  chfnrn  5648  isbasis3g  14023  eltg2b  14031  tgcl  14041  epttop  14067  txuni2  14233
  Copyright terms: Public domain W3C validator