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Theorem sbcbr2g 4091
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 4089 . 2  |-  ( A  e.  D  ->  ( [. A  /  x ]. B R C  <->  [_ A  /  x ]_ B R [_ A  /  x ]_ C
) )
2 csbconstg 3108 . . 3  |-  ( A  e.  D  ->  [_ A  /  x ]_ B  =  B )
32breq1d 4049 . 2  |-  ( A  e.  D  ->  ( [_ A  /  x ]_ B R [_ A  /  x ]_ C  <->  B R [_ A  /  x ]_ C ) )
41, 3bitrd 244 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 176    e. wcel 1696   [.wsbc 3004   [_csb 3094   class class class wbr 4039
This theorem is referenced by:  bnj110  29206
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-br 4040
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