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Theorem intexabim 3935
Description: The intersection of an inhabited class abstraction exists. (Contributed by Jim Kingdon, 27-Aug-2018.)
Assertion
Ref Expression
intexabim (∃𝑥𝜑 {𝑥𝜑} ∈ V)

Proof of Theorem intexabim
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 abid 2070 . . 3 (𝑥 ∈ {𝑥𝜑} ↔ 𝜑)
21exbii 1537 . 2 (∃𝑥 𝑥 ∈ {𝑥𝜑} ↔ ∃𝑥𝜑)
3 nfsab1 2072 . . . 4 𝑥 𝑦 ∈ {𝑥𝜑}
4 nfv 1462 . . . 4 𝑦 𝑥 ∈ {𝑥𝜑}
5 eleq1 2142 . . . 4 (𝑦 = 𝑥 → (𝑦 ∈ {𝑥𝜑} ↔ 𝑥 ∈ {𝑥𝜑}))
63, 4, 5cbvex 1680 . . 3 (∃𝑦 𝑦 ∈ {𝑥𝜑} ↔ ∃𝑥 𝑥 ∈ {𝑥𝜑})
7 inteximm 3932 . . 3 (∃𝑦 𝑦 ∈ {𝑥𝜑} → {𝑥𝜑} ∈ V)
86, 7sylbir 133 . 2 (∃𝑥 𝑥 ∈ {𝑥𝜑} → {𝑥𝜑} ∈ V)
92, 8sylbir 133 1 (∃𝑥𝜑 {𝑥𝜑} ∈ V)
Colors of variables: wff set class
Syntax hints:  wi 4  wex 1422  wcel 1434  {cab 2068  Vcvv 2602   cint 3644
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3904
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-v 2604  df-in 2980  df-ss 2987  df-int 3645
This theorem is referenced by:  intexrabim  3936  omex  4342
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