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Theorem sbccsb2g 3110
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 2271 . . 3  |-  ( x  e.  { x  | 
ph }  <->  ph )
21sbcbii 3046 . 2  |-  ( [. A  /  x ]. x  e.  { x  |  ph } 
<-> 
[. A  /  x ]. ph )
3 sbcel12g 3096 . . 3  |-  ( A  e.  V  ->  ( [. A  /  x ]. x  e.  { x  |  ph }  <->  [_ A  /  x ]_ x  e.  [_ A  /  x ]_ {
x  |  ph }
) )
4 csbvarg 3108 . . . 4  |-  ( A  e.  V  ->  [_ A  /  x ]_ x  =  A )
54eleq1d 2349 . . 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 1684   {cab 2269   [.wsbc 2991   [_csb 3081
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-v 2790  df-sbc 2992  df-csb 3082
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