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Theorem intexrabim 3904
 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 3903 . 2 (∃𝑥(𝑥𝐴𝜑) → {𝑥 ∣ (𝑥𝐴𝜑)} ∈ V)
2 df-rex 2309 . 2 (∃𝑥𝐴 𝜑 ↔ ∃𝑥(𝑥𝐴𝜑))
3 df-rab 2312 . . . 4 {𝑥𝐴𝜑} = {𝑥 ∣ (𝑥𝐴𝜑)}
43inteqi 3616 . . 3 {𝑥𝐴𝜑} = {𝑥 ∣ (𝑥𝐴𝜑)}
54eleq1i 2103 . 2 ( {𝑥𝐴𝜑} ∈ V ↔ {𝑥 ∣ (𝑥𝐴𝜑)} ∈ V)
61, 2, 53imtr4i 190 1 (∃𝑥𝐴 𝜑 {𝑥𝐴𝜑} ∈ V)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97  ∃wex 1381   ∈ wcel 1393  {cab 2026  ∃wrex 2304  {crab 2307  Vcvv 2554  ∩ cint 3612 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3872 This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2308  df-rex 2309  df-rab 2312  df-v 2556  df-in 2921  df-ss 2928  df-int 3613 This theorem is referenced by:  cardcl  6318  isnumi  6319  cardval3ex  6322
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