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Theorem sbcbr2g 4264
Description: Move substitution in and out of a binary relation. (Contributed by NM, 13-Dec-2005.)
Assertion
Ref Expression
sbcbr2g  |-  ( A  e.  D  ->  ( [. A  /  x ]. B R C  <->  B R [_ A  /  x ]_ C ) )
Distinct variable groups:    x, B    x, R
Allowed substitution hints:    A( x)    C( x)    D( x)

Proof of Theorem sbcbr2g
StepHypRef Expression
1 sbcbr12g 4262 . 2  |-  ( A  e.  D  ->  ( [. A  /  x ]. B R C  <->  [_ A  /  x ]_ B R [_ A  /  x ]_ C
) )
2 csbconstg 3265 . . 3  |-  ( A  e.  D  ->  [_ A  /  x ]_ B  =  B )
32breq1d 4222 . 2  |-  ( A  e.  D  ->  ( [_ A  /  x ]_ B R [_ A  /  x ]_ C  <->  B R [_ A  /  x ]_ C ) )
41, 3bitrd 245 1  |-  ( A  e.  D  ->  ( [. A  /  x ]. B R C  <->  B R [_ A  /  x ]_ C ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    e. wcel 1725   [.wsbc 3161   [_csb 3251   class class class wbr 4212
This theorem is referenced by:  bnj110  29229
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-br 4213
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