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Theorem sbccsb2g 3144
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 2304 . . 3  |-  ( x  e.  { x  | 
ph }  <->  ph )
21sbcbii 3080 . 2  |-  ( [. A  /  x ]. x  e.  { x  |  ph } 
<-> 
[. A  /  x ]. ph )
3 sbcel12g 3130 . . 3  |-  ( A  e.  V  ->  ( [. A  /  x ]. x  e.  { x  |  ph }  <->  [_ A  /  x ]_ x  e.  [_ A  /  x ]_ {
x  |  ph }
) )
4 csbvarg 3142 . . . 4  |-  ( A  e.  V  ->  [_ A  /  x ]_ x  =  A )
54eleq1d 2382 . . 3  |-  ( A  e.  V  ->  ( [_ A  /  x ]_ x  e.  [_ A  /  x ]_ { x  |  ph }  <->  A  e.  [_ A  /  x ]_ { x  |  ph }
) )
63, 5bitrd 244 . 2  |-  ( A  e.  V  ->  ( [. A  /  x ]. x  e.  { x  |  ph }  <->  A  e.  [_ A  /  x ]_ { x  |  ph }
) )
72, 6syl5bbr 250 1  |-  ( A  e.  V  ->  ( [. A  /  x ]. ph  <->  A  e.  [_ A  /  x ]_ { x  |  ph } ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    e. wcel 1701   {cab 2302   [.wsbc 3025   [_csb 3115
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-v 2824  df-sbc 3026  df-csb 3116
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