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Theorem sbccsb2 3856
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 3346 . 2  |-  ( [. A  /  x ]. ph  ->  A  e.  _V )
2 elex 3127 . 2  |-  ( A  e.  [_ A  /  x ]_ { x  | 
ph }  ->  A  e.  _V )
3 abid 2454 . . . 4  |-  ( x  e.  { x  | 
ph }  <->  ph )
43sbcbii 3396 . . 3  |-  ( [. A  /  x ]. x  e.  { x  |  ph } 
<-> 
[. A  /  x ]. ph )
5 sbcel12 3828 . . . 4  |-  ( [. A  /  x ]. x  e.  { x  |  ph } 
<-> 
[_ A  /  x ]_ x  e.  [_ A  /  x ]_ { x  |  ph } )
6 csbvarg 3853 . . . . 5  |-  ( A  e.  _V  ->  [_ A  /  x ]_ x  =  A )
76eleq1d 2536 . . . 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 1767   {cab 2452   _Vcvv 3118   [.wsbc 3336   [_csb 3440
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-in 3488  df-ss 3495  df-nul 3791
This theorem is referenced by: (None)
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