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Theorem cbvrab 2549
Description: Rule to change the bound variable in a restricted class abstraction, using implicit substitution. This version has bound-variable hypotheses in place of distinct variable conditions. (Contributed by Andrew Salmon, 11-Jul-2011.) (Revised by Mario Carneiro, 9-Oct-2016.)
Hypotheses
Ref Expression
cbvrab.1  F/_
cbvrab.2  F/_
cbvrab.3  F/
cbvrab.4  F/
cbvrab.5
Assertion
Ref Expression
cbvrab  {  |  }  {  |  }

Proof of Theorem cbvrab
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 nfv 1418 . . . 4  F/
2 cbvrab.1 . . . . . 6  F/_
32nfcri 2169 . . . . 5  F/
4 nfs1v 1812 . . . . 5  F/
53, 4nfan 1454 . . . 4  F/
6 eleq1 2097 . . . . 5
7 sbequ12 1651 . . . . 5
86, 7anbi12d 442 . . . 4
91, 5, 8cbvab 2157 . . 3  {  |  }  {  |  }
10 cbvrab.2 . . . . . 6  F/_
1110nfcri 2169 . . . . 5  F/
12 cbvrab.3 . . . . . 6  F/
1312nfsb 1819 . . . . 5  F/
1411, 13nfan 1454 . . . 4  F/
15 nfv 1418 . . . 4  F/
16 eleq1 2097 . . . . 5
17 sbequ 1718 . . . . . 6
18 cbvrab.4 . . . . . . 7  F/
19 cbvrab.5 . . . . . . 7
2018, 19sbie 1671 . . . . . 6
2117, 20syl6bb 185 . . . . 5
2216, 21anbi12d 442 . . . 4
2314, 15, 22cbvab 2157 . . 3  {  |  }  {  |  }
249, 23eqtri 2057 . 2  {  |  }  {  |  }
25 df-rab 2309 . 2  {  |  }  {  |  }
26 df-rab 2309 . 2  {  |  }  {  |  }
2724, 25, 263eqtr4i 2067 1  {  |  }  {  |  }
Colors of variables: wff set class
Syntax hints:   wi 4   wa 97   wb 98   wceq 1242   F/wnf 1346   wcel 1390  wsb 1642   {cab 2023   F/_wnfc 2162   {crab 2304
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-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-rab 2309
This theorem is referenced by:  cbvrabv  2550  elrabsf  2795  tfis  4249
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