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Theorem eluniab 3619
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 3610 . 2 (𝐴 {𝑥𝜑} ↔ ∃𝑦(𝐴𝑦𝑦 ∈ {𝑥𝜑}))
2 nfv 1437 . . . 4 𝑥 𝐴𝑦
3 nfsab1 2046 . . . 4 𝑥 𝑦 ∈ {𝑥𝜑}
42, 3nfan 1473 . . 3 𝑥(𝐴𝑦𝑦 ∈ {𝑥𝜑})
5 nfv 1437 . . 3 𝑦(𝐴𝑥𝜑)
6 eleq2 2117 . . . 4 (𝑦 = 𝑥 → (𝐴𝑦𝐴𝑥))
7 eleq1 2116 . . . . 5 (𝑦 = 𝑥 → (𝑦 ∈ {𝑥𝜑} ↔ 𝑥 ∈ {𝑥𝜑}))
8 abid 2044 . . . . 5 (𝑥 ∈ {𝑥𝜑} ↔ 𝜑)
97, 8syl6bb 189 . . . 4 (𝑦 = 𝑥 → (𝑦 ∈ {𝑥𝜑} ↔ 𝜑))
106, 9anbi12d 450 . . 3 (𝑦 = 𝑥 → ((𝐴𝑦𝑦 ∈ {𝑥𝜑}) ↔ (𝐴𝑥𝜑)))
114, 5, 10cbvex 1655 . 2 (∃𝑦(𝐴𝑦𝑦 ∈ {𝑥𝜑}) ↔ ∃𝑥(𝐴𝑥𝜑))
121, 11bitri 177 1 (𝐴 {𝑥𝜑} ↔ ∃𝑥(𝐴𝑥𝜑))
Colors of variables: wff set class
Syntax hints:  wa 101  wb 102  wex 1397  wcel 1409  {cab 2042   cuni 3607
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-v 2576  df-uni 3608
This theorem is referenced by:  elunirab  3620  dfiun2g  3716  inuni  3936  snnex  4208  elfv  5203  unielxp  5827  tfrlem9  5965  tfr0  5967
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