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Theorem elintab 4958
Description: Membership in the intersection of a class abstraction. (Contributed by NM, 30-Aug-1993.)
Hypothesis
Ref Expression
elintab.ex 𝐴 ∈ V
Assertion
Ref Expression
elintab (𝐴 {𝑥𝜑} ↔ ∀𝑥(𝜑𝐴𝑥))
Distinct variable group:   𝑥,𝐴
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem elintab
StepHypRef Expression
1 elintab.ex . 2 𝐴 ∈ V
2 elintabg 4957 . 2 (𝐴 ∈ V → (𝐴 {𝑥𝜑} ↔ ∀𝑥(𝜑𝐴𝑥)))
31, 2ax-mp 5 1 (𝐴 {𝑥𝜑} ↔ ∀𝑥(𝜑𝐴𝑥))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wal 1538  wcel 2108  {cab 2714  Vcvv 3480   cint 4946
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-ral 3062  df-int 4947
This theorem is referenced by:  elintrab  4960  intmin4  4977  intab  4978  intidOLD  5463  dfom3  9687  dfom5  9690  tc2  9782  dfnn2  12279  brintclab  15040  efgi  19737  efgi2  19743  dfn0s2  28336  mclsax  35574  heibor1lem  37816  intabssd  43532  elmapintab  43609  cotrintab  43627  dffrege76  43952
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