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Theorem sbccsb2g 2960
Description: Substitution into a wff expressed in using substitution into a class. (Contributed by NM, 27-Nov-2005.)
Assertion
Ref Expression
sbccsb2g  |-  ( A  e.  V  ->  ( [. A  /  x ]. ph  <->  A  e.  [_ A  /  x ]_ { x  |  ph } ) )

Proof of Theorem sbccsb2g
StepHypRef Expression
1 abid 2076 . . 3  |-  ( x  e.  { x  | 
ph }  <->  ph )
21sbcbii 2898 . 2  |-  ( [. A  /  x ]. x  e.  { x  |  ph } 
<-> 
[. A  /  x ]. ph )
3 sbcel12g 2946 . . 3  |-  ( A  e.  V  ->  ( [. A  /  x ]. x  e.  { x  |  ph }  <->  [_ A  /  x ]_ x  e.  [_ A  /  x ]_ {
x  |  ph }
) )
4 csbvarg 2958 . . . 4  |-  ( A  e.  V  ->  [_ A  /  x ]_ x  =  A )
54eleq1d 2156 . . 3  |-  ( A  e.  V  ->  ( [_ A  /  x ]_ x  e.  [_ A  /  x ]_ { x  |  ph }  <->  A  e.  [_ A  /  x ]_ { x  |  ph }
) )
63, 5bitrd 186 . 2  |-  ( A  e.  V  ->  ( [. A  /  x ]. x  e.  { x  |  ph }  <->  A  e.  [_ A  /  x ]_ { x  |  ph }
) )
72, 6syl5bbr 192 1  |-  ( A  e.  V  ->  ( [. A  /  x ]. ph  <->  A  e.  [_ A  /  x ]_ { x  |  ph } ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 103    e. wcel 1438   {cab 2074   [.wsbc 2840   [_csb 2933
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-sbc 2841  df-csb 2934
This theorem is referenced by: (None)
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