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Theorem dfuni2 4909
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 4908 . 2 𝐴 = {𝑥 ∣ ∃𝑦(𝑥𝑦𝑦𝐴)}
2 exancom 1861 . . . 4 (∃𝑦(𝑥𝑦𝑦𝐴) ↔ ∃𝑦(𝑦𝐴𝑥𝑦))
3 df-rex 3071 . . . 4 (∃𝑦𝐴 𝑥𝑦 ↔ ∃𝑦(𝑦𝐴𝑥𝑦))
42, 3bitr4i 278 . . 3 (∃𝑦(𝑥𝑦𝑦𝐴) ↔ ∃𝑦𝐴 𝑥𝑦)
54abbii 2809 . 2 {𝑥 ∣ ∃𝑦(𝑥𝑦𝑦𝐴)} = {𝑥 ∣ ∃𝑦𝐴 𝑥𝑦}
61, 5eqtri 2765 1 𝐴 = {𝑥 ∣ ∃𝑦𝐴 𝑥𝑦}
Colors of variables: wff setvar class
Syntax hints:  wa 395   = wceq 1540  wex 1779  wcel 2108  {cab 2714  wrex 3070   cuni 4907
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-rex 3071  df-uni 4908
This theorem is referenced by:  nfunid  4913  uniiun  5058  rncnvepres  38304  uniel  43229  onsupmaxb  43251
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