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Theorem sbccsb2 3833
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 3321 . 2  |-  ( [. A  /  x ]. ph  ->  A  e.  _V )
2 elex 3102 . 2  |-  ( A  e.  [_ A  /  x ]_ { x  | 
ph }  ->  A  e.  _V )
3 abid 2428 . . . 4  |-  ( x  e.  { x  | 
ph }  <->  ph )
43sbcbii 3371 . . 3  |-  ( [. A  /  x ]. x  e.  { x  |  ph } 
<-> 
[. A  /  x ]. ph )
5 sbcel12 3805 . . . 4  |-  ( [. A  /  x ]. x  e.  { x  |  ph } 
<-> 
[_ A  /  x ]_ x  e.  [_ A  /  x ]_ { x  |  ph } )
6 csbvarg 3830 . . . . 5  |-  ( A  e.  _V  ->  [_ A  /  x ]_ x  =  A )
76eleq1d 2510 . . . 4  |-  ( A  e.  _V  ->  ( [_ A  /  x ]_ x  e.  [_ A  /  x ]_ { x  |  ph }  <->  A  e.  [_ A  /  x ]_ { x  |  ph }
) )
85, 7syl5bb 257 . . 3  |-  ( A  e.  _V  ->  ( [. A  /  x ]. x  e.  { x  |  ph }  <->  A  e.  [_ A  /  x ]_ { x  |  ph }
) )
94, 8syl5bbr 259 . 2  |-  ( A  e.  _V  ->  ( [. A  /  x ]. ph  <->  A  e.  [_ A  /  x ]_ { x  |  ph } ) )
101, 2, 9pm5.21nii 353 1  |-  ( [. A  /  x ]. ph  <->  A  e.  [_ A  /  x ]_ { x  |  ph }
)
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    e. wcel 1802   {cab 2426   _Vcvv 3093   [.wsbc 3311   [_csb 3417
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1384  df-fal 1387  df-ex 1598  df-nf 1602  df-sb 1725  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-v 3095  df-sbc 3312  df-csb 3418  df-dif 3461  df-in 3465  df-ss 3472  df-nul 3768
This theorem is referenced by: (None)
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