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Theorem eluniab 4921
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 4910 . 2 (𝐴 {𝑥𝜑} ↔ ∃𝑦(𝐴𝑦𝑦 ∈ {𝑥𝜑}))
2 nfv 1914 . . . 4 𝑥 𝐴𝑦
3 nfsab1 2722 . . . 4 𝑥 𝑦 ∈ {𝑥𝜑}
42, 3nfan 1899 . . 3 𝑥(𝐴𝑦𝑦 ∈ {𝑥𝜑})
5 nfv 1914 . . 3 𝑦(𝐴𝑥𝜑)
6 eleq2w 2825 . . . 4 (𝑦 = 𝑥 → (𝐴𝑦𝐴𝑥))
7 eleq1w 2824 . . . . 5 (𝑦 = 𝑥 → (𝑦 ∈ {𝑥𝜑} ↔ 𝑥 ∈ {𝑥𝜑}))
8 abid 2718 . . . . 5 (𝑥 ∈ {𝑥𝜑} ↔ 𝜑)
97, 8bitrdi 287 . . . 4 (𝑦 = 𝑥 → (𝑦 ∈ {𝑥𝜑} ↔ 𝜑))
106, 9anbi12d 632 . . 3 (𝑦 = 𝑥 → ((𝐴𝑦𝑦 ∈ {𝑥𝜑}) ↔ (𝐴𝑥𝜑)))
114, 5, 10cbvexv1 2344 . 2 (∃𝑦(𝐴𝑦𝑦 ∈ {𝑥𝜑}) ↔ ∃𝑥(𝐴𝑥𝜑))
121, 11bitri 275 1 (𝐴 {𝑥𝜑} ↔ ∃𝑥(𝐴𝑥𝜑))
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395  wex 1779  wcel 2108  {cab 2714   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-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1543  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-v 3482  df-uni 4908
This theorem is referenced by:  elunirab  4922  dfiun2gOLD  5031  inuni  5350  elfv  6904  unielxp  8052  frrlem8  8318  frrlem10  8320  wfrlem12OLD  8360  tfrlem9  8425  dfac5lem2  10164  fin23lem30  10382  unisngl  23535  metrest  24537  aannenlem2  26371  fpwrelmapffslem  32743  dfiota3  35924  mptsnunlem  37339  nnoeomeqom  43325
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