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Theorem intexabim 3903
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 2028 . . 3 (𝑥 ∈ {𝑥𝜑} ↔ 𝜑)
21exbii 1496 . 2 (∃𝑥 𝑥 ∈ {𝑥𝜑} ↔ ∃𝑥𝜑)
3 nfsab1 2030 . . . 4 𝑥 𝑦 ∈ {𝑥𝜑}
4 nfv 1421 . . . 4 𝑦 𝑥 ∈ {𝑥𝜑}
5 eleq1 2100 . . . 4 (𝑦 = 𝑥 → (𝑦 ∈ {𝑥𝜑} ↔ 𝑥 ∈ {𝑥𝜑}))
63, 4, 5cbvex 1639 . . 3 (∃𝑦 𝑦 ∈ {𝑥𝜑} ↔ ∃𝑥 𝑥 ∈ {𝑥𝜑})
7 inteximm 3900 . . 3 (∃𝑦 𝑦 ∈ {𝑥𝜑} → {𝑥𝜑} ∈ V)
86, 7sylbir 125 . 2 (∃𝑥 𝑥 ∈ {𝑥𝜑} → {𝑥𝜑} ∈ V)
92, 8sylbir 125 1 (∃𝑥𝜑 {𝑥𝜑} ∈ V)
Colors of variables: wff set class
Syntax hints:  wi 4  wex 1381  wcel 1393  {cab 2026  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-v 2556  df-in 2921  df-ss 2928  df-int 3613
This theorem is referenced by:  intexrabim  3904  omex  4279
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