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Theorem sbcbrg 3804
Description: Move substitution in and out of a binary relation. (Contributed by NM, 13-Dec-2005.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
Assertion
Ref Expression
sbcbrg  D  [.  ]. R C  [_  ]_ [_  ]_ R [_  ]_ C

Proof of Theorem sbcbrg
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 dfsbcq2 2761 . 2  R C  [.  ]. R C
2 csbeq1 2849 . . 3  [_  ]_  [_  ]_
3 csbeq1 2849 . . 3  [_  ]_ R 
[_  ]_ R
4 csbeq1 2849 . . 3  [_  ]_ C 
[_  ]_ C
52, 3, 4breq123d 3769 . 2  [_  ]_ [_  ]_ R [_  ]_ C  [_  ]_ [_  ]_ R [_  ]_ C
6 nfcsb1v 2876 . . . 4  F/_ [_  ]_
7 nfcsb1v 2876 . . . 4  F/_ [_  ]_ R
8 nfcsb1v 2876 . . . 4  F/_ [_  ]_ C
96, 7, 8nfbr 3799 . . 3  F/ [_  ]_ [_  ]_ R [_  ]_ C
10 csbeq1a 2854 . . . 4  [_  ]_
11 csbeq1a 2854 . . . 4  R  [_  ]_ R
12 csbeq1a 2854 . . . 4  C  [_  ]_ C
1310, 11, 12breq123d 3769 . . 3  R C  [_  ]_ [_  ]_ R [_  ]_ C
149, 13sbie 1671 . 2  R C  [_  ]_ [_  ]_ R [_  ]_ C
151, 5, 14vtoclbg 2608 1  D  [.  ]. R C  [_  ]_ [_  ]_ R [_  ]_ C
Colors of variables: wff set class
Syntax hints:   wi 4   wb 98   wceq 1242   wcel 1390  wsb 1642   [.wsbc 2758   [_csb 2846   class class class wbr 3755
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-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  df-un 2916  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756
This theorem is referenced by:  sbcbr12g  3805  csbcnvg  4462  sbcfung  4868  csbfv12g  5152
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