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Theorem sbcbr1g 4176
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 4175 . 2  |-  ( A  e.  D  ->  ( [. A  /  x ]. B R C  <->  [_ A  /  x ]_ B R [_ A  /  x ]_ C
) )
2 csbconstg 3181 . . 3  |-  ( A  e.  D  ->  [_ A  /  x ]_ C  =  C )
32breq2d 4137 . 2  |-  ( A  e.  D  ->  ( [_ A  /  x ]_ B R [_ A  /  x ]_ C  <->  [_ A  /  x ]_ B R C ) )
41, 3bitrd 244 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 176    e. wcel 1715   [.wsbc 3077   [_csb 3167   class class class wbr 4125
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-rab 2637  df-v 2875  df-sbc 3078  df-csb 3168  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-sn 3735  df-pr 3736  df-op 3738  df-br 4126
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