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Theorem dfuni2 4625
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 4624 . 2 𝐴 = {𝑥 ∣ ∃𝑦(𝑥𝑦𝑦𝐴)}
2 exancom 1947 . . . 4 (∃𝑦(𝑥𝑦𝑦𝐴) ↔ ∃𝑦(𝑦𝐴𝑥𝑦))
3 df-rex 3098 . . . 4 (∃𝑦𝐴 𝑥𝑦 ↔ ∃𝑦(𝑦𝐴𝑥𝑦))
42, 3bitr4i 269 . . 3 (∃𝑦(𝑥𝑦𝑦𝐴) ↔ ∃𝑦𝐴 𝑥𝑦)
54abbii 2919 . 2 {𝑥 ∣ ∃𝑦(𝑥𝑦𝑦𝐴)} = {𝑥 ∣ ∃𝑦𝐴 𝑥𝑦}
61, 5eqtri 2824 1 𝐴 = {𝑥 ∣ ∃𝑦𝐴 𝑥𝑦}
Colors of variables: wff setvar class
Syntax hints:  wa 384   = wceq 1637  wex 1859  wcel 2155  {cab 2788  wrex 3093   cuni 4623
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2067  ax-7 2103  ax-9 2164  ax-10 2184  ax-11 2200  ax-12 2213  ax-ext 2781
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2060  df-clab 2789  df-cleq 2795  df-clel 2798  df-rex 3098  df-uni 4624
This theorem is referenced by:  nfuni  4629  nfunid  4630  unieq  4631  uniiun  4758  rncnvepres  34383
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