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Theorem intexrabim 3898
Description: The intersection of an inhabited restricted class abstraction exists. (Contributed by Jim Kingdon, 27-Aug-2018.)
Assertion
Ref Expression
intexrabim (x A φ {x Aφ} V)

Proof of Theorem intexrabim
StepHypRef Expression
1 intexabim 3897 . 2 (x(x A φ) → {x ∣ (x A φ)} V)
2 df-rex 2306 . 2 (x A φx(x A φ))
3 df-rab 2309 . . . 4 {x Aφ} = {x ∣ (x A φ)}
43inteqi 3610 . . 3 {x Aφ} = {x ∣ (x A φ)}
54eleq1i 2100 . 2 ( {x Aφ} V ↔ {x ∣ (x A φ)} V)
61, 2, 53imtr4i 190 1 (x A φ {x Aφ} V)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wex 1378   wcel 1390  {cab 2023  wrex 2301  {crab 2304  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-bnd 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-ral 2305  df-rex 2306  df-rab 2309  df-v 2553  df-in 2918  df-ss 2925  df-int 3607
This theorem is referenced by: (None)
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