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Theorem spc3gv 2639
Description: Specialization with 3 quantifiers, using implicit substitution. (Contributed by NM, 12-May-2008.)
Hypothesis
Ref Expression
spc3egv.1  C
Assertion
Ref Expression
spc3gv  V  W  C  X
Distinct variable groups:   ,,,   ,,,   , C,,   ,,,
Allowed substitution hints:   (,,)    V(,,)    W(,,)    X(,,)

Proof of Theorem spc3gv
StepHypRef Expression
1 elisset 2562 . . . 4  V
2 elisset 2562 . . . 4  W
3 elisset 2562 . . . 4  C  X  C
41, 2, 33anim123i 1088 . . 3  V  W  C  X  C
5 eeeanv 1805 . . 3  C  C
64, 5sylibr 137 . 2  V  W  C  X  C
7 spc3egv.1 . . . . . . . 8  C
87biimpcd 148 . . . . . . 7  C
982alimi 1342 . . . . . 6  C
109alimi 1341 . . . . 5  C
11 exim 1487 . . . . . 6  C  C
12112alimi 1342 . . . . 5  C  C
1310, 12syl 14 . . . 4  C
14 exim 1487 . . . . 5  C  C
1514alimi 1341 . . . 4  C  C
16 exim 1487 . . . 4  C  C
1713, 15, 163syl 17 . . 3  C
18 19.9v 1748 . . . 4
19 19.9v 1748 . . . 4
20 19.9v 1748 . . . 4
2118, 19, 203bitri 195 . . 3
2217, 21syl6ib 150 . 2  C
236, 22syl5com 26 1  V  W  C  X
Colors of variables: wff set class
Syntax hints:   wi 4   wb 98   w3a 884  wal 1240   wceq 1242  wex 1378   wcel 1390
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:  funopg  4877
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