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Theorem sbceq2g 3791
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 3785 . 2  |-  ( A  e.  V  ->  ( [. A  /  x ]. B  =  C  <->  [_ A  /  x ]_ B  =  [_ A  /  x ]_ C ) )
2 csbconstg 3388 . . 3  |-  ( A  e.  V  ->  [_ A  /  x ]_ B  =  B )
32eqeq1d 2464 . 2  |-  ( A  e.  V  ->  ( [_ A  /  x ]_ B  =  [_ A  /  x ]_ C  <->  B  =  [_ A  /  x ]_ C ) )
41, 3bitrd 261 1  |-  ( A  e.  V  ->  ( [. A  /  x ]. B  =  C  <->  B  =  [_ A  /  x ]_ C ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    = wceq 1455    e. wcel 1898   [.wsbc 3279   [_csb 3375
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-v 3059  df-sbc 3280  df-csb 3376
This theorem is referenced by:  csbsng  4042  csbmpt12  4749  f1od2  28358  bj-snsetex  31602  csbmpt22g  31777  csbfinxpg  31825  poimirlem26  32011  cdlemkid3N  34545  cdlemkid4  34546  brtrclfv2  36364  frege116  36620
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