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Theorem csbiebt 2880
Description: Conversion of implicit substitution to explicit substitution into a class. (Closed theorem version of csbiegf 2884.) (Contributed by NM, 11-Nov-2005.)
Assertion
Ref Expression
csbiebt  V  F/_ C  C  [_  ]_  C
Distinct variable group:   ,
Allowed substitution hints:   ()    C()    V()

Proof of Theorem csbiebt
StepHypRef Expression
1 elex 2560 . 2  V  _V
2 spsbc 2769 . . . . 5  _V  C  [.  ].  C
32adantr 261 . . . 4  _V  F/_ C  C  [.  ].  C
4 simpl 102 . . . . 5  _V  F/_ C  _V
5 biimt 230 . . . . . . 7  C  C
6 csbeq1a 2854 . . . . . . . 8  [_  ]_
76eqeq1d 2045 . . . . . . 7  C  [_  ]_  C
85, 7bitr3d 179 . . . . . 6  C 
[_  ]_  C
98adantl 262 . . . . 5  _V 
F/_ C  C  [_  ]_  C
10 nfv 1418 . . . . . 6  F/  _V
11 nfnfc1 2178 . . . . . 6  F/ F/_ C
1210, 11nfan 1454 . . . . 5  F/  _V 
F/_ C
13 nfcsb1v 2876 . . . . . . 7  F/_ [_  ]_
1413a1i 9 . . . . . 6  _V  F/_ C  F/_ [_  ]_
15 simpr 103 . . . . . 6  _V  F/_ C  F/_ C
1614, 15nfeqd 2189 . . . . 5  _V  F/_ C 
F/ [_  ]_  C
174, 9, 12, 16sbciedf 2792 . . . 4  _V  F/_ C  [.  ].  C 
[_  ]_  C
183, 17sylibd 138 . . 3  _V  F/_ C  C  [_  ]_  C
1913a1i 9 . . . . . . . 8  F/_ C  F/_ [_  ]_
20 id 19 . . . . . . . 8  F/_ C  F/_ C
2119, 20nfeqd 2189 . . . . . . 7  F/_ C  F/ [_  ]_  C
2211, 21nfan1 1453 . . . . . 6  F/ F/_ C  [_  ]_  C
237biimprcd 149 . . . . . . 7  [_  ]_  C  C
2423adantl 262 . . . . . 6 
F/_ C  [_  ]_  C  C
2522, 24alrimi 1412 . . . . 5 
F/_ C  [_  ]_  C  C
2625ex 108 . . . 4  F/_ C  [_  ]_  C  C
2726adantl 262 . . 3  _V  F/_ C  [_  ]_  C  C
2818, 27impbid 120 . 2  _V  F/_ C  C  [_  ]_  C
291, 28sylan 267 1  V  F/_ C  C  [_  ]_  C
Colors of variables: wff set class
Syntax hints:   wi 4   wa 97   wb 98  wal 1240   wceq 1242   wcel 1390   F/_wnfc 2162   _Vcvv 2551   [.wsbc 2758   [_csb 2846
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
This theorem depends on definitions:  df-bi 110  df-3an 886  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-csb 2847
This theorem is referenced by:  csbiedf  2881  csbieb  2882  csbiegf  2884
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