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Theorem sbccsb2 3761
Description: Substitution into a wff expressed in using substitution into a class. (Contributed by NM, 27-Nov-2005.) (Revised by NM, 18-Aug-2018.)
Assertion
Ref Expression
sbccsb2  |-  ( [. A  /  x ]. ph  <->  A  e.  [_ A  /  x ]_ { x  |  ph }
)

Proof of Theorem sbccsb2
StepHypRef Expression
1 sbcex 3244 . 2  |-  ( [. A  /  x ]. ph  ->  A  e.  _V )
2 elex 3021 . 2  |-  ( A  e.  [_ A  /  x ]_ { x  | 
ph }  ->  A  e.  _V )
3 abid 2439 . . . 4  |-  ( x  e.  { x  | 
ph }  <->  ph )
43sbcbii 3290 . . 3  |-  ( [. A  /  x ]. x  e.  { x  |  ph } 
<-> 
[. A  /  x ]. ph )
5 sbcel12 3739 . . . 4  |-  ( [. A  /  x ]. x  e.  { x  |  ph } 
<-> 
[_ A  /  x ]_ x  e.  [_ A  /  x ]_ { x  |  ph } )
6 csbvarg 3759 . . . . 5  |-  ( A  e.  _V  ->  [_ A  /  x ]_ x  =  A )
76eleq1d 2513 . . . 4  |-  ( A  e.  _V  ->  ( [_ A  /  x ]_ x  e.  [_ A  /  x ]_ { x  |  ph }  <->  A  e.  [_ A  /  x ]_ { x  |  ph }
) )
85, 7syl5bb 265 . . 3  |-  ( A  e.  _V  ->  ( [. A  /  x ]. x  e.  { x  |  ph }  <->  A  e.  [_ A  /  x ]_ { x  |  ph }
) )
94, 8syl5bbr 267 . 2  |-  ( A  e.  _V  ->  ( [. A  /  x ]. ph  <->  A  e.  [_ A  /  x ]_ { x  |  ph } ) )
101, 2, 9pm5.21nii 359 1  |-  ( [. A  /  x ]. ph  <->  A  e.  [_ A  /  x ]_ { x  |  ph }
)
Colors of variables: wff setvar class
Syntax hints:    <-> wb 189    e. wcel 1890   {cab 2437   _Vcvv 3012   [.wsbc 3234   [_csb 3330
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1672  ax-4 1685  ax-5 1761  ax-6 1808  ax-7 1854  ax-10 1918  ax-11 1923  ax-12 1936  ax-13 2091  ax-ext 2431
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-tru 1450  df-fal 1453  df-ex 1667  df-nf 1671  df-sb 1801  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-v 3014  df-sbc 3235  df-csb 3331  df-dif 3374  df-in 3378  df-ss 3385  df-nul 3699
This theorem is referenced by: (None)
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