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Theorem intab 3635
Description: The intersection of a special case of a class abstraction. may be free in and , which can be thought of a and . (Contributed by NM, 28-Jul-2006.) (Proof shortened by Mario Carneiro, 14-Nov-2016.)
Hypotheses
Ref Expression
intab.1  _V
intab.2  {  |  }  _V
Assertion
Ref Expression
intab  |^| {  |  }  {  |  }
Distinct variable groups:   ,   ,   ,
Allowed substitution hints:   ()   ()

Proof of Theorem intab
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 eqeq1 2043 . . . . . . . . . 10
21anbi2d 437 . . . . . . . . 9
32exbidv 1703 . . . . . . . 8
43cbvabv 2158 . . . . . . 7  {  |  }  {  |  }
5 intab.2 . . . . . . 7  {  |  }  _V
64, 5eqeltri 2107 . . . . . 6  {  |  }  _V
7 nfe1 1382 . . . . . . . . 9  F/
87nfab 2179 . . . . . . . 8  F/_ {  |  }
98nfeq2 2186 . . . . . . 7  F/  {  |  }
10 eleq2 2098 . . . . . . . 8  {  |  }  {  |  }
1110imbi2d 219 . . . . . . 7  {  |  }  {  |  }
129, 11albid 1503 . . . . . 6  {  |  }  {  |  }
136, 12elab 2681 . . . . 5  {  |  }  {  |  }  {  |  }
14 19.8a 1479 . . . . . . . . 9
1514ex 108 . . . . . . . 8
1615alrimiv 1751 . . . . . . 7
17 intab.1 . . . . . . . 8  _V
1817sbc6 2783 . . . . . . 7  [.  ].
1916, 18sylibr 137 . . . . . 6  [.  ].
20 df-sbc 2759 . . . . . 6  [.  ].  {  |  }
2119, 20sylib 127 . . . . 5  {  |  }
2213, 21mpgbir 1339 . . . 4  {  |  }  {  |  }
23 intss1 3621 . . . 4  {  |  }  {  |  }  |^| {  |  }  C_  {  |  }
2422, 23ax-mp 7 . . 3  |^| {  |  }  C_  {  |  }
25 19.29r 1509 . . . . . . . 8
26 simplr 482 . . . . . . . . . 10
27 pm3.35 329 . . . . . . . . . . 11
2827adantlr 446 . . . . . . . . . 10
2926, 28eqeltrd 2111 . . . . . . . . 9
3029exlimiv 1486 . . . . . . . 8
3125, 30syl 14 . . . . . . 7
3231ex 108 . . . . . 6
3332alrimiv 1751 . . . . 5
34 vex 2554 . . . . . 6 
_V
3534elintab 3617 . . . . 5  |^| {  |  }
3633, 35sylibr 137 . . . 4  |^| {  |  }
3736abssi 3009 . . 3  {  |  }  C_  |^| {  |  }
3824, 37eqssi 2955 . 2  |^| {  |  }  {  |  }
3938, 4eqtri 2057 1  |^| {  |  }  {  |  }
Colors of variables: wff set class
Syntax hints:   wi 4   wa 97  wal 1240   wceq 1242  wex 1378   wcel 1390   {cab 2023   _Vcvv 2551   [.wsbc 2758    C_ wss 2911   |^|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
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-sbc 2759  df-in 2918  df-ss 2925  df-int 3607
This theorem is referenced by: (None)
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