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Theorem sbcbr1g 4227
Description: Move substitution in and out of a binary relation. (Contributed by NM, 13-Dec-2005.)
Assertion
Ref Expression
sbcbr1g  |-  ( A  e.  D  ->  ( [. A  /  x ]. B R C  <->  [_ A  /  x ]_ B R C ) )
Distinct variable groups:    x, C    x, R
Allowed substitution hints:    A( x)    B( x)    D( x)

Proof of Theorem sbcbr1g
StepHypRef Expression
1 sbcbr12g 4226 . 2  |-  ( A  e.  D  ->  ( [. A  /  x ]. B R C  <->  [_ A  /  x ]_ B R [_ A  /  x ]_ C
) )
2 csbconstg 3229 . . 3  |-  ( A  e.  D  ->  [_ A  /  x ]_ C  =  C )
32breq2d 4188 . 2  |-  ( A  e.  D  ->  ( [_ A  /  x ]_ B R [_ A  /  x ]_ C  <->  [_ A  /  x ]_ B R C ) )
41, 3bitrd 245 1  |-  ( A  e.  D  ->  ( [. A  /  x ]. B R C  <->  [_ A  /  x ]_ B R C ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    e. wcel 1721   [.wsbc 3125   [_csb 3215   class class class wbr 4176
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-rab 2679  df-v 2922  df-sbc 3126  df-csb 3216  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-nul 3593  df-if 3704  df-sn 3784  df-pr 3785  df-op 3787  df-br 4177
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