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Theorem sbceqg 2860
Description: Distribute proper substitution through an equality relation. (Contributed by NM, 10-Nov-2005.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
Assertion
Ref Expression
sbceqg  V  [.  ].  C  [_  ]_  [_  ]_ C

Proof of Theorem sbceqg
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dfsbcq2 2761 . . 3  C  [.  ].  C
2 dfsbcq2 2761 . . . . 5  [.  ].
32abbidv 2152 . . . 4  {  |  }  {  |  [.  ].  }
4 dfsbcq2 2761 . . . . 5  C  [.  ].  C
54abbidv 2152 . . . 4  {  |  C }  {  |  [.  ].  C }
63, 5eqeq12d 2051 . . 3  {  |  }  {  |  C }  {  |  [.  ].  }  {  |  [.  ].  C }
7 nfs1v 1812 . . . . . 6  F/
87nfab 2179 . . . . 5  F/_ {  |  }
9 nfs1v 1812 . . . . . 6  F/  C
109nfab 2179 . . . . 5  F/_ {  |  C }
118, 10nfeq 2182 . . . 4  F/ {  |  }  {  |  C }
12 sbab 2161 . . . . 5  {  |  }
13 sbab 2161 . . . . 5  C  {  |  C }
1412, 13eqeq12d 2051 . . . 4  C  {  |  }  {  |  C }
1511, 14sbie 1671 . . 3  C  {  |  }  {  |  C }
161, 6, 15vtoclbg 2608 . 2  V  [.  ].  C  {  |  [.  ].  }  {  |  [.  ].  C }
17 df-csb 2847 . . 3  [_  ]_  {  |  [.  ].  }
18 df-csb 2847 . . 3  [_  ]_ C  {  |  [.  ].  C }
1917, 18eqeq12i 2050 . 2  [_  ]_  [_  ]_ C  {  | 
[.  ].  }  {  |  [.  ].  C }
2016, 19syl6bbr 187 1  V  [.  ].  C  [_  ]_  [_  ]_ C
Colors of variables: wff set class
Syntax hints:   wi 4   wb 98   wceq 1242   wcel 1390  wsb 1642   {cab 2023   [.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-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:  sbcne12g  2862  sbceq1g  2864  sbceq2g  2866  sbcfng  4987
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