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Theorem intexrabim 4048
Description: The intersection of an inhabited restricted class abstraction exists. (Contributed by Jim Kingdon, 27-Aug-2018.)
Assertion
Ref Expression
intexrabim (∃𝑥𝐴 𝜑 {𝑥𝐴𝜑} ∈ V)

Proof of Theorem intexrabim
StepHypRef Expression
1 intexabim 4047 . 2 (∃𝑥(𝑥𝐴𝜑) → {𝑥 ∣ (𝑥𝐴𝜑)} ∈ V)
2 df-rex 2399 . 2 (∃𝑥𝐴 𝜑 ↔ ∃𝑥(𝑥𝐴𝜑))
3 df-rab 2402 . . . 4 {𝑥𝐴𝜑} = {𝑥 ∣ (𝑥𝐴𝜑)}
43inteqi 3745 . . 3 {𝑥𝐴𝜑} = {𝑥 ∣ (𝑥𝐴𝜑)}
54eleq1i 2183 . 2 ( {𝑥𝐴𝜑} ∈ V ↔ {𝑥 ∣ (𝑥𝐴𝜑)} ∈ V)
61, 2, 53imtr4i 200 1 (∃𝑥𝐴 𝜑 {𝑥𝐴𝜑} ∈ V)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wex 1453  wcel 1465  {cab 2103  wrex 2394  {crab 2397  Vcvv 2660   cint 3741
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-rab 2402  df-v 2662  df-in 3047  df-ss 3054  df-int 3742
This theorem is referenced by:  cardcl  7005  isnumi  7006  cardval3ex  7009  clsval  12207
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