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Theorem sbcbr1g 4043
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 4042 . 2  |-  ( A  e.  D  ->  ( [. A  /  x ]. B R C  <->  [_ A  /  x ]_ B R [_ A  /  x ]_ C
) )
2 csbconstg 3063 . . 3  |-  ( A  e.  D  ->  [_ A  /  x ]_ C  =  C )
32breq2d 3999 . 2  |-  ( A  e.  D  ->  ( [_ A  /  x ]_ B R [_ A  /  x ]_ C  <->  [_ A  /  x ]_ B R C ) )
41, 3bitrd 187 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 104    e. wcel 2141   [.wsbc 2955   [_csb 3049   class class class wbr 3987
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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-sbc 2956  df-csb 3050  df-un 3125  df-sn 3587  df-pr 3588  df-op 3590  df-br 3988
This theorem is referenced by: (None)
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