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Theorem sbceq2g 3241
Description: Move proper substitution to second argument of an equality. (Contributed by NM, 30-Nov-2005.)
Assertion
Ref Expression
sbceq2g  |-  ( A  e.  V  ->  ( [. A  /  x ]. B  =  C  <->  B  =  [_ A  /  x ]_ C ) )
Distinct variable group:    x, B
Allowed substitution hints:    A( x)    C( x)    V( x)

Proof of Theorem sbceq2g
StepHypRef Expression
1 sbceqg 3235 . 2  |-  ( A  e.  V  ->  ( [. A  /  x ]. B  =  C  <->  [_ A  /  x ]_ B  =  [_ A  /  x ]_ C ) )
2 csbconstg 3233 . . 3  |-  ( A  e.  V  ->  [_ A  /  x ]_ B  =  B )
32eqeq1d 2420 . 2  |-  ( A  e.  V  ->  ( [_ A  /  x ]_ B  =  [_ A  /  x ]_ C  <->  B  =  [_ A  /  x ]_ C ) )
41, 3bitrd 245 1  |-  ( A  e.  V  ->  ( [. A  /  x ]. B  =  C  <->  B  =  [_ A  /  x ]_ C ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    = wceq 1649    e. wcel 1721   [.wsbc 3129   [_csb 3219
This theorem is referenced by:  csbsng  3835  cdlemkid3N  31427  cdlemkid4  31428
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 2393
This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-v 2926  df-sbc 3130  df-csb 3220
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