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Theorem sbcbrg 3942
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  |-  ( A  e.  D  ->  ( [. A  /  x ]. B R C  <->  [_ A  /  x ]_ B [_ A  /  x ]_ R [_ A  /  x ]_ C
) )

Proof of Theorem sbcbrg
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 dfsbcq2 2879 . 2  |-  ( y  =  A  ->  ( [ y  /  x ] B R C  <->  [. A  /  x ]. B R C ) )
2 csbeq1 2972 . . 3  |-  ( y  =  A  ->  [_ y  /  x ]_ B  = 
[_ A  /  x ]_ B )
3 csbeq1 2972 . . 3  |-  ( y  =  A  ->  [_ y  /  x ]_ R  = 
[_ A  /  x ]_ R )
4 csbeq1 2972 . . 3  |-  ( y  =  A  ->  [_ y  /  x ]_ C  = 
[_ A  /  x ]_ C )
52, 3, 4breq123d 3907 . 2  |-  ( y  =  A  ->  ( [_ y  /  x ]_ B [_ y  /  x ]_ R [_ y  /  x ]_ C  <->  [_ A  /  x ]_ B [_ A  /  x ]_ R [_ A  /  x ]_ C
) )
6 nfcsb1v 2999 . . . 4  |-  F/_ x [_ y  /  x ]_ B
7 nfcsb1v 2999 . . . 4  |-  F/_ x [_ y  /  x ]_ R
8 nfcsb1v 2999 . . . 4  |-  F/_ x [_ y  /  x ]_ C
96, 7, 8nfbr 3937 . . 3  |-  F/ x [_ y  /  x ]_ B [_ y  /  x ]_ R [_ y  /  x ]_ C
10 csbeq1a 2977 . . . 4  |-  ( x  =  y  ->  B  =  [_ y  /  x ]_ B )
11 csbeq1a 2977 . . . 4  |-  ( x  =  y  ->  R  =  [_ y  /  x ]_ R )
12 csbeq1a 2977 . . . 4  |-  ( x  =  y  ->  C  =  [_ y  /  x ]_ C )
1310, 11, 12breq123d 3907 . . 3  |-  ( x  =  y  ->  ( B R C  <->  [_ y  /  x ]_ B [_ y  /  x ]_ R [_ y  /  x ]_ C
) )
149, 13sbie 1745 . 2  |-  ( [ y  /  x ] B R C  <->  [_ y  /  x ]_ B [_ y  /  x ]_ R [_ y  /  x ]_ C
)
151, 5, 14vtoclbg 2716 1  |-  ( A  e.  D  ->  ( [. A  /  x ]. B R C  <->  [_ A  /  x ]_ B [_ A  /  x ]_ R [_ A  /  x ]_ C
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104    = wceq 1312    e. wcel 1461   [wsb 1716   [.wsbc 2876   [_csb 2969   class class class wbr 3893
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-10 1464  ax-11 1465  ax-i12 1466  ax-bndl 1467  ax-4 1468  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095
This theorem depends on definitions:  df-bi 116  df-3an 945  df-tru 1315  df-nf 1418  df-sb 1717  df-clab 2100  df-cleq 2106  df-clel 2109  df-nfc 2242  df-v 2657  df-sbc 2877  df-csb 2970  df-un 3039  df-sn 3497  df-pr 3498  df-op 3500  df-br 3894
This theorem is referenced by:  sbcbr12g  3943  csbcnvg  4681  sbcfung  5103  csbfv12g  5409
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