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Theorem eluniab 4923
Description: Membership in union of a class abstraction. (Contributed by NM, 11-Aug-1994.) (Revised by Mario Carneiro, 14-Nov-2016.)
Assertion
Ref Expression
eluniab (𝐴 {𝑥𝜑} ↔ ∃𝑥(𝐴𝑥𝜑))
Distinct variable group:   𝑥,𝐴
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem eluniab
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 eluni 4911 . 2 (𝐴 {𝑥𝜑} ↔ ∃𝑦(𝐴𝑦𝑦 ∈ {𝑥𝜑}))
2 nfv 1917 . . . 4 𝑥 𝐴𝑦
3 nfsab1 2717 . . . 4 𝑥 𝑦 ∈ {𝑥𝜑}
42, 3nfan 1902 . . 3 𝑥(𝐴𝑦𝑦 ∈ {𝑥𝜑})
5 nfv 1917 . . 3 𝑦(𝐴𝑥𝜑)
6 eleq2w 2817 . . . 4 (𝑦 = 𝑥 → (𝐴𝑦𝐴𝑥))
7 eleq1w 2816 . . . . 5 (𝑦 = 𝑥 → (𝑦 ∈ {𝑥𝜑} ↔ 𝑥 ∈ {𝑥𝜑}))
8 abid 2713 . . . . 5 (𝑥 ∈ {𝑥𝜑} ↔ 𝜑)
97, 8bitrdi 286 . . . 4 (𝑦 = 𝑥 → (𝑦 ∈ {𝑥𝜑} ↔ 𝜑))
106, 9anbi12d 631 . . 3 (𝑦 = 𝑥 → ((𝐴𝑦𝑦 ∈ {𝑥𝜑}) ↔ (𝐴𝑥𝜑)))
114, 5, 10cbvexv1 2338 . 2 (∃𝑦(𝐴𝑦𝑦 ∈ {𝑥𝜑}) ↔ ∃𝑥(𝐴𝑥𝜑))
121, 11bitri 274 1 (𝐴 {𝑥𝜑} ↔ ∃𝑥(𝐴𝑥𝜑))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 396  wex 1781  wcel 2106  {cab 2709   cuni 4908
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-tru 1544  df-ex 1782  df-nf 1786  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-v 3476  df-uni 4909
This theorem is referenced by:  elunirab  4924  dfiun2gOLD  5034  inuni  5343  elfv  6889  unielxp  8015  frrlem8  8280  frrlem10  8282  wfrlem12OLD  8322  tfrlem9  8387  dfac5lem2  10121  fin23lem30  10339  unisngl  23251  metrest  24253  aannenlem2  26066  fpwrelmapffslem  32212  dfiota3  35187  mptsnunlem  36522  nnoeomeqom  42364
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