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Theorem dfuni2 3823
Description: Alternate definition of class union. (Contributed by NM, 28-Jun-1998.)
Assertion
Ref Expression
dfuni2 𝐴 = {𝑥 ∣ ∃𝑦𝐴 𝑥𝑦}
Distinct variable group:   𝑥,𝑦,𝐴

Proof of Theorem dfuni2
StepHypRef Expression
1 df-uni 3822 . 2 𝐴 = {𝑥 ∣ ∃𝑦(𝑥𝑦𝑦𝐴)}
2 exancom 1618 . . . 4 (∃𝑦(𝑥𝑦𝑦𝐴) ↔ ∃𝑦(𝑦𝐴𝑥𝑦))
3 df-rex 2471 . . . 4 (∃𝑦𝐴 𝑥𝑦 ↔ ∃𝑦(𝑦𝐴𝑥𝑦))
42, 3bitr4i 187 . . 3 (∃𝑦(𝑥𝑦𝑦𝐴) ↔ ∃𝑦𝐴 𝑥𝑦)
54abbii 2303 . 2 {𝑥 ∣ ∃𝑦(𝑥𝑦𝑦𝐴)} = {𝑥 ∣ ∃𝑦𝐴 𝑥𝑦}
61, 5eqtri 2208 1 𝐴 = {𝑥 ∣ ∃𝑦𝐴 𝑥𝑦}
Colors of variables: wff set class
Syntax hints:  wa 104   = wceq 1363  wex 1502  wcel 2158  {cab 2173  wrex 2466   cuni 3821
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-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-11 1516  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169
This theorem depends on definitions:  df-bi 117  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-rex 2471  df-uni 3822
This theorem is referenced by:  nfuni  3827  nfunid  3828  unieq  3830  uniiun  3952
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