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Theorem sbcbr12g 3953
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 3952 . 2  |-  ( A  e.  D  ->  ( [. A  /  x ]. B R C  <->  [_ A  /  x ]_ B [_ A  /  x ]_ R [_ A  /  x ]_ C
) )
2 csbconstg 2987 . . 3  |-  ( A  e.  D  ->  [_ A  /  x ]_ R  =  R )
32breqd 3910 . 2  |-  ( A  e.  D  ->  ( [_ A  /  x ]_ B [_ A  /  x ]_ R [_ A  /  x ]_ C  <->  [_ A  /  x ]_ B R [_ A  /  x ]_ C
) )
41, 3bitrd 187 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 104    e. wcel 1465   [.wsbc 2882   [_csb 2975   class class class wbr 3899
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-v 2662  df-sbc 2883  df-csb 2976  df-un 3045  df-sn 3503  df-pr 3504  df-op 3506  df-br 3900
This theorem is referenced by:  sbcbr1g  3954  sbcbr2g  3955
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