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Theorem elintab 4452
Description: Membership in the intersection of a class abstraction. (Contributed by NM, 30-Aug-1993.)
Hypothesis
Ref Expression
inteqab.1 𝐴 ∈ V
Assertion
Ref Expression
elintab (𝐴 {𝑥𝜑} ↔ ∀𝑥(𝜑𝐴𝑥))
Distinct variable group:   𝑥,𝐴
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem elintab
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 inteqab.1 . . 3 𝐴 ∈ V
21elint 4446 . 2 (𝐴 {𝑥𝜑} ↔ ∀𝑦(𝑦 ∈ {𝑥𝜑} → 𝐴𝑦))
3 nfsab1 2611 . . . 4 𝑥 𝑦 ∈ {𝑥𝜑}
4 nfv 1840 . . . 4 𝑥 𝐴𝑦
53, 4nfim 1822 . . 3 𝑥(𝑦 ∈ {𝑥𝜑} → 𝐴𝑦)
6 nfv 1840 . . 3 𝑦(𝜑𝐴𝑥)
7 eleq1 2686 . . . . 5 (𝑦 = 𝑥 → (𝑦 ∈ {𝑥𝜑} ↔ 𝑥 ∈ {𝑥𝜑}))
8 abid 2609 . . . . 5 (𝑥 ∈ {𝑥𝜑} ↔ 𝜑)
97, 8syl6bb 276 . . . 4 (𝑦 = 𝑥 → (𝑦 ∈ {𝑥𝜑} ↔ 𝜑))
10 eleq2 2687 . . . 4 (𝑦 = 𝑥 → (𝐴𝑦𝐴𝑥))
119, 10imbi12d 334 . . 3 (𝑦 = 𝑥 → ((𝑦 ∈ {𝑥𝜑} → 𝐴𝑦) ↔ (𝜑𝐴𝑥)))
125, 6, 11cbval 2270 . 2 (∀𝑦(𝑦 ∈ {𝑥𝜑} → 𝐴𝑦) ↔ ∀𝑥(𝜑𝐴𝑥))
132, 12bitri 264 1 (𝐴 {𝑥𝜑} ↔ ∀𝑥(𝜑𝐴𝑥))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wal 1478  wcel 1987  {cab 2607  Vcvv 3186   cint 4440
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-v 3188  df-int 4441
This theorem is referenced by:  elintrab  4453  intmin4  4471  intab  4472  intid  4887  dfom3  8488  dfom5  8491  tc2  8562  dfnn2  10977  brintclab  13676  efgi  18053  efgi2  18059  mclsax  31174  heibor1lem  33240  elmapintab  37383  intabssd  37397  cotrintab  37402  dffrege76  37715
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