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Theorem sbcbr12g 4222
Description: Move substitution in and out of a binary relation. (Contributed by NM, 13-Dec-2005.)
Assertion
Ref Expression
sbcbr12g  |-  ( A  e.  D  ->  ( [. A  /  x ]. B R C  <->  [_ A  /  x ]_ B R [_ A  /  x ]_ C
) )
Distinct variable group:    x, R
Allowed substitution hints:    A( x)    B( x)    C( x)    D( x)

Proof of Theorem sbcbr12g
StepHypRef Expression
1 sbcbrg 4221 . 2  |-  ( A  e.  D  ->  ( [. A  /  x ]. B R C  <->  [_ A  /  x ]_ B [_ A  /  x ]_ R [_ A  /  x ]_ C
) )
2 csbconstg 3225 . . 3  |-  ( A  e.  D  ->  [_ A  /  x ]_ R  =  R )
32breqd 4183 . 2  |-  ( A  e.  D  ->  ( [_ A  /  x ]_ B [_ A  /  x ]_ R [_ A  /  x ]_ C  <->  [_ A  /  x ]_ B R [_ A  /  x ]_ C
) )
41, 3bitrd 245 1  |-  ( A  e.  D  ->  ( [. A  /  x ]. B R C  <->  [_ A  /  x ]_ B R [_ A  /  x ]_ C
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    e. wcel 1721   [.wsbc 3121   [_csb 3211   class class class wbr 4172
This theorem is referenced by:  sbcbr1g  4223  sbcbr2g  4224  cdlemk39s  31421
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 2385
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 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-br 4173
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