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Theorem intexabim 4113
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 2145 . . 3 (𝑥 ∈ {𝑥𝜑} ↔ 𝜑)
21exbii 1585 . 2 (∃𝑥 𝑥 ∈ {𝑥𝜑} ↔ ∃𝑥𝜑)
3 nfsab1 2147 . . . 4 𝑥 𝑦 ∈ {𝑥𝜑}
4 nfv 1508 . . . 4 𝑦 𝑥 ∈ {𝑥𝜑}
5 eleq1 2220 . . . 4 (𝑦 = 𝑥 → (𝑦 ∈ {𝑥𝜑} ↔ 𝑥 ∈ {𝑥𝜑}))
63, 4, 5cbvex 1736 . . 3 (∃𝑦 𝑦 ∈ {𝑥𝜑} ↔ ∃𝑥 𝑥 ∈ {𝑥𝜑})
7 inteximm 4110 . . 3 (∃𝑦 𝑦 ∈ {𝑥𝜑} → {𝑥𝜑} ∈ V)
86, 7sylbir 134 . 2 (∃𝑥 𝑥 ∈ {𝑥𝜑} → {𝑥𝜑} ∈ V)
92, 8sylbir 134 1 (∃𝑥𝜑 {𝑥𝜑} ∈ V)
Colors of variables: wff set class
Syntax hints:  wi 4  wex 1472  wcel 2128  {cab 2143  Vcvv 2712   cint 3807
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 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139  ax-sep 4082
This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-v 2714  df-in 3108  df-ss 3115  df-int 3808
This theorem is referenced by:  intexrabim  4114  omex  4552
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