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Theorem sbccsb2g 2945
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 2071 . . 3  |-  ( x  e.  { x  | 
ph }  <->  ph )
21sbcbii 2883 . 2  |-  ( [. A  /  x ]. x  e.  { x  |  ph } 
<-> 
[. A  /  x ]. ph )
3 sbcel12g 2931 . . 3  |-  ( A  e.  V  ->  ( [. A  /  x ]. x  e.  { x  |  ph }  <->  [_ A  /  x ]_ x  e.  [_ A  /  x ]_ {
x  |  ph }
) )
4 csbvarg 2943 . . . 4  |-  ( A  e.  V  ->  [_ A  /  x ]_ x  =  A )
54eleq1d 2151 . . 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 1434   {cab 2069   [.wsbc 2825   [_csb 2918
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 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-v 2612  df-sbc 2826  df-csb 2919
This theorem is referenced by: (None)
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