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Theorem sbcbrg 4072
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 2994 . 2  |-  ( y  =  A  ->  ( [ y  /  x ] B R C  <->  [. A  /  x ]. B R C ) )
2 csbeq1 3084 . . 3  |-  ( y  =  A  ->  [_ y  /  x ]_ B  = 
[_ A  /  x ]_ B )
3 csbeq1 3084 . . 3  |-  ( y  =  A  ->  [_ y  /  x ]_ R  = 
[_ A  /  x ]_ R )
4 csbeq1 3084 . . 3  |-  ( y  =  A  ->  [_ y  /  x ]_ C  = 
[_ A  /  x ]_ C )
52, 3, 4breq123d 4037 . 2  |-  ( y  =  A  ->  ( [_ y  /  x ]_ B [_ y  /  x ]_ R [_ y  /  x ]_ C  <->  [_ A  /  x ]_ B [_ A  /  x ]_ R [_ A  /  x ]_ C
) )
6 nfcsb1v 3113 . . . 4  |-  F/_ x [_ y  /  x ]_ B
7 nfcsb1v 3113 . . . 4  |-  F/_ x [_ y  /  x ]_ R
8 nfcsb1v 3113 . . . 4  |-  F/_ x [_ y  /  x ]_ C
96, 7, 8nfbr 4067 . . 3  |-  F/ x [_ y  /  x ]_ B [_ y  /  x ]_ R [_ y  /  x ]_ C
10 csbeq1a 3089 . . . 4  |-  ( x  =  y  ->  B  =  [_ y  /  x ]_ B )
11 csbeq1a 3089 . . . 4  |-  ( x  =  y  ->  R  =  [_ y  /  x ]_ R )
12 csbeq1a 3089 . . . 4  |-  ( x  =  y  ->  C  =  [_ y  /  x ]_ C )
1310, 11, 12breq123d 4037 . . 3  |-  ( x  =  y  ->  ( B R C  <->  [_ y  /  x ]_ B [_ y  /  x ]_ R [_ y  /  x ]_ C
) )
149, 13sbie 1978 . 2  |-  ( [ y  /  x ] B R C  <->  [_ y  /  x ]_ B [_ y  /  x ]_ R [_ y  /  x ]_ C
)
151, 5, 14vtoclbg 2844 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 176    = wceq 1623   [wsb 1629    e. wcel 1684   [.wsbc 2991   [_csb 3081   class class class wbr 4023
This theorem is referenced by:  sbcbr12g  4073  csbfv12g  5535  csbcnvg  23198  sbcfun  27985
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024
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