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Theorem eluni2 3612
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 1515 . 2 (∃𝑥(𝐴𝑥𝑥𝐵) ↔ ∃𝑥(𝑥𝐵𝐴𝑥))
2 eluni 3611 . 2 (𝐴 𝐵 ↔ ∃𝑥(𝐴𝑥𝑥𝐵))
3 df-rex 2329 . 2 (∃𝑥𝐵 𝐴𝑥 ↔ ∃𝑥(𝑥𝐵𝐴𝑥))
41, 2, 33bitr4i 205 1 (𝐴 𝐵 ↔ ∃𝑥𝐵 𝐴𝑥)
Colors of variables: wff set class
Syntax hints:  wa 101  wb 102  wex 1397  wcel 1409  wrex 2324   cuni 3608
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-rex 2329  df-v 2576  df-uni 3609
This theorem is referenced by:  uni0b  3633  intssunim  3665  iuncom4  3692  inuni  3937  ssorduni  4241  unon  4265  cnvuni  4549  chfnrn  5306
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