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Theorem intexabim 3897
 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 2025 . . 3 (x {xφ} ↔ φ)
21exbii 1493 . 2 (x x {xφ} ↔ xφ)
3 nfsab1 2027 . . . 4 x y {xφ}
4 nfv 1418 . . . 4 y x {xφ}
5 eleq1 2097 . . . 4 (y = x → (y {xφ} ↔ x {xφ}))
63, 4, 5cbvex 1636 . . 3 (y y {xφ} ↔ x x {xφ})
7 inteximm 3894 . . 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 1378   ∈ wcel 1390  {cab 2023  Vcvv 2551  ∩ cint 3606 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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866 This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-in 2918  df-ss 2925  df-int 3607 This theorem is referenced by:  intexrabim  3898  omex  4259
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