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Theorem csbab 3825
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 2408 . . . 4  |-  ( z  e.  { y  | 
[. A  /  x ]. ph }  <->  [ z  /  y ] [. A  /  x ]. ph )
2 sbsbc 3303 . . . 4  |-  ( [ z  /  y ]
[. A  /  x ]. ph  <->  [. z  /  y ]. [. A  /  x ]. ph )
31, 2bitri 252 . . 3  |-  ( z  e.  { y  | 
[. A  /  x ]. ph }  <->  [. z  / 
y ]. [. A  /  x ]. ph )
4 sbccom 3371 . . . 4  |-  ( [. z  /  y ]. [. A  /  x ]. ph  <->  [. A  /  x ]. [. z  / 
y ]. ph )
5 df-clab 2408 . . . . . 6  |-  ( z  e.  { y  | 
ph }  <->  [ z  /  y ] ph )
6 sbsbc 3303 . . . . . 6  |-  ( [ z  /  y ]
ph 
<-> 
[. z  /  y ]. ph )
75, 6bitri 252 . . . . 5  |-  ( z  e.  { y  | 
ph }  <->  [. z  / 
y ]. ph )
87sbcbii 3355 . . . 4  |-  ( [. A  /  x ]. z  e.  { y  |  ph } 
<-> 
[. A  /  x ]. [. z  /  y ]. ph )
94, 8bitr4i 255 . . 3  |-  ( [. z  /  y ]. [. A  /  x ]. ph  <->  [. A  /  x ]. z  e.  {
y  |  ph }
)
10 sbcel2 3806 . . 3  |-  ( [. A  /  x ]. z  e.  { y  |  ph } 
<->  z  e.  [_ A  /  x ]_ { y  |  ph } )
113, 9, 103bitrri 275 . 2  |-  ( z  e.  [_ A  /  x ]_ { y  | 
ph }  <->  z  e.  { y  |  [. A  /  x ]. ph }
)
1211eqriv 2418 1  |-  [_ A  /  x ]_ { y  |  ph }  =  { y  |  [. A  /  x ]. ph }
Colors of variables: wff setvar class
Syntax hints:    = wceq 1437   [wsb 1786    e. wcel 1868   {cab 2407   [.wsbc 3299   [_csb 3395
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-tru 1440  df-fal 1443  df-ex 1660  df-nf 1664  df-sb 1787  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-v 3083  df-sbc 3300  df-csb 3396  df-dif 3439  df-in 3443  df-ss 3450  df-nul 3762
This theorem is referenced by:  csbsng  4055  csbuni  4244  csbxp  4932  csbdm  5045  csbwrdg  12689  abfmpeld  28243  abfmpel  28244
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