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Theorem cbvralcsf 2902
Description: A more general version of cbvralf 2521 that doesn't require and to be distinct from or . Changes bound variables using implicit substitution. (Contributed by Andrew Salmon, 13-Jul-2011.)
Hypotheses
Ref Expression
cbvralcsf.1  F/_
cbvralcsf.2  F/_
cbvralcsf.3  F/
cbvralcsf.4  F/
cbvralcsf.5
cbvralcsf.6
Assertion
Ref Expression
cbvralcsf

Proof of Theorem cbvralcsf
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nfv 1418 . . . 4  F/
2 nfcsb1v 2876 . . . . . 6  F/_ [_  ]_
32nfcri 2169 . . . . 5  F/  [_  ]_
4 nfsbc1v 2776 . . . . 5  F/ [.  ].
53, 4nfim 1461 . . . 4  F/  [_  ]_  [.  ].
6 id 19 . . . . . 6
7 csbeq1a 2854 . . . . . 6  [_  ]_
86, 7eleq12d 2105 . . . . 5 
[_  ]_
9 sbceq1a 2767 . . . . 5 
[.  ].
108, 9imbi12d 223 . . . 4  [_  ]_  [.  ].
111, 5, 10cbval 1634 . . 3  [_  ]_  [.  ].
12 nfcv 2175 . . . . . . 7  F/_
13 cbvralcsf.1 . . . . . . 7  F/_
1412, 13nfcsb 2878 . . . . . 6  F/_ [_  ]_
1514nfcri 2169 . . . . 5  F/  [_  ]_
16 cbvralcsf.3 . . . . . 6  F/
1712, 16nfsbc 2778 . . . . 5  F/
[.  ].
1815, 17nfim 1461 . . . 4  F/  [_  ]_  [.  ].
19 nfv 1418 . . . 4  F/
20 id 19 . . . . . 6
21 csbeq1 2849 . . . . . . 7  [_  ]_  [_  ]_
22 df-csb 2847 . . . . . . . 8  [_  ]_  {  |  [.  ].  }
23 cbvralcsf.2 . . . . . . . . . . . 12  F/_
2423nfcri 2169 . . . . . . . . . . 11  F/
25 cbvralcsf.5 . . . . . . . . . . . 12
2625eleq2d 2104 . . . . . . . . . . 11
2724, 26sbie 1671 . . . . . . . . . 10
28 sbsbc 2762 . . . . . . . . . 10  [.  ].
2927, 28bitr3i 175 . . . . . . . . 9  [.  ].
3029abbi2i 2149 . . . . . . . 8  {  |  [.  ].  }
3122, 30eqtr4i 2060 . . . . . . 7  [_  ]_
3221, 31syl6eq 2085 . . . . . 6  [_  ]_
3320, 32eleq12d 2105 . . . . 5  [_  ]_
34 dfsbcq 2760 . . . . . 6  [.  ]. 
[.  ].
35 sbsbc 2762 . . . . . . 7  [.  ].
36 cbvralcsf.4 . . . . . . . 8  F/
37 cbvralcsf.6 . . . . . . . 8
3836, 37sbie 1671 . . . . . . 7
3935, 38bitr3i 175 . . . . . 6  [.  ].
4034, 39syl6bb 185 . . . . 5  [.  ].
4133, 40imbi12d 223 . . . 4  [_  ]_  [.  ].
4218, 19, 41cbval 1634 . . 3  [_  ]_  [.  ].
4311, 42bitri 173 . 2
44 df-ral 2305 . 2
45 df-ral 2305 . 2
4643, 44, 453bitr4i 201 1
Colors of variables: wff set class
Syntax hints:   wi 4   wb 98  wal 1240   wceq 1242   F/wnf 1346   wcel 1390  wsb 1642   {cab 2023   F/_wnfc 2162  wral 2300   [.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-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-ral 2305  df-sbc 2759  df-csb 2847
This theorem is referenced by:  cbvralv2  2906
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