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Theorem dfuni2 4374
 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 4373 . 2 𝐴 = {𝑥 ∣ ∃𝑦(𝑥𝑦𝑦𝐴)}
2 exancom 1774 . . . 4 (∃𝑦(𝑥𝑦𝑦𝐴) ↔ ∃𝑦(𝑦𝐴𝑥𝑦))
3 df-rex 2902 . . . 4 (∃𝑦𝐴 𝑥𝑦 ↔ ∃𝑦(𝑦𝐴𝑥𝑦))
42, 3bitr4i 266 . . 3 (∃𝑦(𝑥𝑦𝑦𝐴) ↔ ∃𝑦𝐴 𝑥𝑦)
54abbii 2726 . 2 {𝑥 ∣ ∃𝑦(𝑥𝑦𝑦𝐴)} = {𝑥 ∣ ∃𝑦𝐴 𝑥𝑦}
61, 5eqtri 2632 1 𝐴 = {𝑥 ∣ ∃𝑦𝐴 𝑥𝑦}
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 383   = wceq 1475  ∃wex 1695   ∈ wcel 1977  {cab 2596  ∃wrex 2897  ∪ cuni 4372 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-rex 2902  df-uni 4373 This theorem is referenced by:  nfuni  4378  nfunid  4379  unieq  4380  uniiun  4509
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