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Theorem csbabg 3155
Description: Move substitution into a class abstraction. (Contributed by NM, 13-Dec-2005.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
Assertion
Ref Expression
csbabg  |-  ( A  e.  V  ->  [_ A  /  x ]_ { y  |  ph }  =  { y  |  [. A  /  x ]. ph }
)
Distinct variable groups:    y, A    x, y
Allowed substitution hints:    ph( x, y)    A( x)    V( x, y)

Proof of Theorem csbabg
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 sbccom 3075 . . . 4  |-  ( [. z  /  y ]. [. A  /  x ]. ph  <->  [. A  /  x ]. [. z  / 
y ]. ph )
2 df-clab 2283 . . . . 5  |-  ( z  e.  { y  | 
[. A  /  x ]. ph }  <->  [ z  /  y ] [. A  /  x ]. ph )
3 sbsbc 3008 . . . . 5  |-  ( [ z  /  y ]
[. A  /  x ]. ph  <->  [. z  /  y ]. [. A  /  x ]. ph )
42, 3bitri 240 . . . 4  |-  ( z  e.  { y  | 
[. A  /  x ]. ph }  <->  [. z  / 
y ]. [. A  /  x ]. ph )
5 df-clab 2283 . . . . . 6  |-  ( z  e.  { y  | 
ph }  <->  [ z  /  y ] ph )
6 sbsbc 3008 . . . . . 6  |-  ( [ z  /  y ]
ph 
<-> 
[. z  /  y ]. ph )
75, 6bitri 240 . . . . 5  |-  ( z  e.  { y  | 
ph }  <->  [. z  / 
y ]. ph )
87sbcbii 3059 . . . 4  |-  ( [. A  /  x ]. z  e.  { y  |  ph } 
<-> 
[. A  /  x ]. [. z  /  y ]. ph )
91, 4, 83bitr4i 268 . . 3  |-  ( z  e.  { y  | 
[. A  /  x ]. ph }  <->  [. A  /  x ]. z  e.  {
y  |  ph }
)
10 sbcel2g 3115 . . 3  |-  ( A  e.  V  ->  ( [. A  /  x ]. z  e.  { y  |  ph }  <->  z  e.  [_ A  /  x ]_ { y  |  ph } ) )
119, 10syl5rbb 249 . 2  |-  ( A  e.  V  ->  (
z  e.  [_ A  /  x ]_ { y  |  ph }  <->  z  e.  { y  |  [. A  /  x ]. ph }
) )
1211eqrdv 2294 1  |-  ( A  e.  V  ->  [_ A  /  x ]_ { y  |  ph }  =  { y  |  [. A  /  x ]. ph }
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1632   [wsb 1638    e. wcel 1696   {cab 2282   [.wsbc 3004   [_csb 3094
This theorem is referenced by:  csbsng  3705  csbunig  3851  csbxpg  4732  csbrng  4939  csbfv12gALT  5552  abfmpeld  23233  abfmpel  23234  csbdmg  28085  csbingVD  28976  csbsngVD  28985  csbxpgVD  28986  csbrngVD  28988  csbunigVD  28990  csbfv12gALTVD  28991
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-v 2803  df-sbc 3005  df-csb 3095
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