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Theorem intexabim 3956
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 2073 . . 3 (𝑥 ∈ {𝑥𝜑} ↔ 𝜑)
21exbii 1539 . 2 (∃𝑥 𝑥 ∈ {𝑥𝜑} ↔ ∃𝑥𝜑)
3 nfsab1 2075 . . . 4 𝑥 𝑦 ∈ {𝑥𝜑}
4 nfv 1464 . . . 4 𝑦 𝑥 ∈ {𝑥𝜑}
5 eleq1 2147 . . . 4 (𝑦 = 𝑥 → (𝑦 ∈ {𝑥𝜑} ↔ 𝑥 ∈ {𝑥𝜑}))
63, 4, 5cbvex 1683 . . 3 (∃𝑦 𝑦 ∈ {𝑥𝜑} ↔ ∃𝑥 𝑥 ∈ {𝑥𝜑})
7 inteximm 3953 . . 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 1424  wcel 1436  {cab 2071  Vcvv 2614   cint 3665
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 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-sep 3925
This theorem depends on definitions:  df-bi 115  df-tru 1290  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-v 2616  df-in 2992  df-ss 2999  df-int 3666
This theorem is referenced by:  intexrabim  3957  omex  4374
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