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Theorem dfuni2 4409
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 4408 . 2 𝐴 = {𝑥 ∣ ∃𝑦(𝑥𝑦𝑦𝐴)}
2 exancom 1784 . . . 4 (∃𝑦(𝑥𝑦𝑦𝐴) ↔ ∃𝑦(𝑦𝐴𝑥𝑦))
3 df-rex 2913 . . . 4 (∃𝑦𝐴 𝑥𝑦 ↔ ∃𝑦(𝑦𝐴𝑥𝑦))
42, 3bitr4i 267 . . 3 (∃𝑦(𝑥𝑦𝑦𝐴) ↔ ∃𝑦𝐴 𝑥𝑦)
54abbii 2736 . 2 {𝑥 ∣ ∃𝑦(𝑥𝑦𝑦𝐴)} = {𝑥 ∣ ∃𝑦𝐴 𝑥𝑦}
61, 5eqtri 2643 1 𝐴 = {𝑥 ∣ ∃𝑦𝐴 𝑥𝑦}
Colors of variables: wff setvar class
Syntax hints:  wa 384   = wceq 1480  wex 1701  wcel 1987  {cab 2607  wrex 2908   cuni 4407
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-rex 2913  df-uni 4408
This theorem is referenced by:  nfuni  4413  nfunid  4414  unieq  4415  uniiun  4544
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