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Theorem sbcbr2g 4479
Description: Move substitution in and out of a binary relation. (Contributed by NM, 13-Dec-2005.)
Assertion
Ref Expression
sbcbr2g  |-  ( A  e.  V  ->  ( [. A  /  x ]. B R C  <->  B R [_ A  /  x ]_ C ) )
Distinct variable groups:    x, B    x, R
Allowed substitution hints:    A( x)    C( x)    V( x)

Proof of Theorem sbcbr2g
StepHypRef Expression
1 sbcbr12g 4477 . 2  |-  ( A  e.  V  ->  ( [. A  /  x ]. B R C  <->  [_ A  /  x ]_ B R [_ A  /  x ]_ C
) )
2 csbconstg 3408 . . 3  |-  ( A  e.  V  ->  [_ A  /  x ]_ B  =  B )
32breq1d 4433 . 2  |-  ( A  e.  V  ->  ( [_ A  /  x ]_ B R [_ A  /  x ]_ C  <->  B R [_ A  /  x ]_ C ) )
41, 3bitrd 256 1  |-  ( A  e.  V  ->  ( [. A  /  x ]. B R C  <->  B R [_ A  /  x ]_ C ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    e. wcel 1872   [.wsbc 3299   [_csb 3395   class class class wbr 4423
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-fal 1443  df-ex 1658  df-nf 1662  df-sb 1791  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-rab 2780  df-v 3082  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3912  df-sn 3999  df-pr 4001  df-op 4005  df-br 4424
This theorem is referenced by:  prmgaplem7  15026  telgsums  17622  fvmptnn04if  19871  bnj110  29677  frege124d  36323  frege72  36501  frege91  36520  frege116  36545  frege120  36549
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