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Theorem csbab 3809
Description: Move substitution into a class abstraction. (Contributed by NM, 13-Dec-2005.) (Revised by NM, 19-Aug-2018.)
Assertion
Ref Expression
csbab  |-  [_ A  /  x ]_ { y  |  ph }  =  { y  |  [. A  /  x ]. ph }
Distinct variable groups:    y, A    x, y
Allowed substitution hints:    ph( x, y)    A( x)

Proof of Theorem csbab
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 df-clab 2449 . . . 4  |-  ( z  e.  { y  | 
[. A  /  x ]. ph }  <->  [ z  /  y ] [. A  /  x ]. ph )
2 sbsbc 3283 . . . 4  |-  ( [ z  /  y ]
[. A  /  x ]. ph  <->  [. z  /  y ]. [. A  /  x ]. ph )
31, 2bitri 257 . . 3  |-  ( z  e.  { y  | 
[. A  /  x ]. ph }  <->  [. z  / 
y ]. [. A  /  x ]. ph )
4 sbccom 3351 . . . 4  |-  ( [. z  /  y ]. [. A  /  x ]. ph  <->  [. A  /  x ]. [. z  / 
y ]. ph )
5 df-clab 2449 . . . . . 6  |-  ( z  e.  { y  | 
ph }  <->  [ z  /  y ] ph )
6 sbsbc 3283 . . . . . 6  |-  ( [ z  /  y ]
ph 
<-> 
[. z  /  y ]. ph )
75, 6bitri 257 . . . . 5  |-  ( z  e.  { y  | 
ph }  <->  [. z  / 
y ]. ph )
87sbcbii 3335 . . . 4  |-  ( [. A  /  x ]. z  e.  { y  |  ph } 
<-> 
[. A  /  x ]. [. z  /  y ]. ph )
94, 8bitr4i 260 . . 3  |-  ( [. z  /  y ]. [. A  /  x ]. ph  <->  [. A  /  x ]. z  e.  {
y  |  ph }
)
10 sbcel2 3790 . . 3  |-  ( [. A  /  x ]. z  e.  { y  |  ph } 
<->  z  e.  [_ A  /  x ]_ { y  |  ph } )
113, 9, 103bitrri 280 . 2  |-  ( z  e.  [_ A  /  x ]_ { y  | 
ph }  <->  z  e.  { y  |  [. A  /  x ]. ph }
)
1211eqriv 2459 1  |-  [_ A  /  x ]_ { y  |  ph }  =  { y  |  [. A  /  x ]. ph }
Colors of variables: wff setvar class
Syntax hints:    = wceq 1455   [wsb 1808    e. wcel 1898   {cab 2448   [.wsbc 3279   [_csb 3375
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-tru 1458  df-fal 1461  df-ex 1675  df-nf 1679  df-sb 1809  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-v 3059  df-sbc 3280  df-csb 3376  df-dif 3419  df-in 3423  df-ss 3430  df-nul 3744
This theorem is referenced by:  csbsng  4042  csbuni  4240  csbxp  4935  csbdm  5048  csbwrdg  12732  abfmpeld  28302  abfmpel  28303  csbwrecsg  31773  csboprabg  31776  csbfinxpg  31825
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