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Theorem ceqsex3v 2590
Description: Elimination of three existential quantifiers, using implicit substitution. (Contributed by NM, 16-Aug-2011.)
Hypotheses
Ref Expression
ceqsex3v.1  _V
ceqsex3v.2  _V
ceqsex3v.3  C 
_V
ceqsex3v.4
ceqsex3v.5
ceqsex3v.6  C
Assertion
Ref Expression
ceqsex3v  C
Distinct variable groups:   ,,,   ,,,   , C,,   ,   ,   ,
Allowed substitution hints:   (,,)   (,)   (,)   (,)

Proof of Theorem ceqsex3v
StepHypRef Expression
1 anass 381 . . . . . 6  C  C
2 3anass 888 . . . . . . 7  C  C
32anbi1i 431 . . . . . 6  C  C
4 df-3an 886 . . . . . . 7  C  C
54anbi2i 430 . . . . . 6  C  C
61, 3, 53bitr4i 201 . . . . 5  C  C
762exbii 1494 . . . 4  C  C
8 19.42vv 1785 . . . 4  C  C
97, 8bitri 173 . . 3  C  C
109exbii 1493 . 2  C  C
11 ceqsex3v.1 . . . 4  _V
12 ceqsex3v.4 . . . . . 6
13123anbi3d 1212 . . . . 5  C  C
14132exbidv 1745 . . . 4  C  C
1511, 14ceqsexv 2587 . . 3  C  C
16 ceqsex3v.2 . . . 4  _V
17 ceqsex3v.3 . . . 4  C 
_V
18 ceqsex3v.5 . . . 4
19 ceqsex3v.6 . . . 4  C
2016, 17, 18, 19ceqsex2v 2589 . . 3  C
2115, 20bitri 173 . 2  C
2210, 21bitri 173 1  C
Colors of variables: wff set class
Syntax hints:   wi 4   wa 97   wb 98   w3a 884   wceq 1242  wex 1378   wcel 1390   _Vcvv 2551
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-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-3an 886  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-v 2553
This theorem is referenced by:  ceqsex6v  2592
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