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Theorem csbabg 3255
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 3177 . . . 4  |-  ( [. z  /  y ]. [. A  /  x ]. ph  <->  [. A  /  x ]. [. z  / 
y ]. ph )
2 df-clab 2376 . . . . 5  |-  ( z  e.  { y  | 
[. A  /  x ]. ph }  <->  [ z  /  y ] [. A  /  x ]. ph )
3 sbsbc 3110 . . . . 5  |-  ( [ z  /  y ]
[. A  /  x ]. ph  <->  [. z  /  y ]. [. A  /  x ]. ph )
42, 3bitri 241 . . . 4  |-  ( z  e.  { y  | 
[. A  /  x ]. ph }  <->  [. z  / 
y ]. [. A  /  x ]. ph )
5 df-clab 2376 . . . . . 6  |-  ( z  e.  { y  | 
ph }  <->  [ z  /  y ] ph )
6 sbsbc 3110 . . . . . 6  |-  ( [ z  /  y ]
ph 
<-> 
[. z  /  y ]. ph )
75, 6bitri 241 . . . . 5  |-  ( z  e.  { y  | 
ph }  <->  [. z  / 
y ]. ph )
87sbcbii 3161 . . . 4  |-  ( [. A  /  x ]. z  e.  { y  |  ph } 
<-> 
[. A  /  x ]. [. z  /  y ]. ph )
91, 4, 83bitr4i 269 . . 3  |-  ( z  e.  { y  | 
[. A  /  x ]. ph }  <->  [. A  /  x ]. z  e.  {
y  |  ph }
)
10 sbcel2g 3217 . . 3  |-  ( A  e.  V  ->  ( [. A  /  x ]. z  e.  { y  |  ph }  <->  z  e.  [_ A  /  x ]_ { y  |  ph } ) )
119, 10syl5rbb 250 . 2  |-  ( A  e.  V  ->  (
z  e.  [_ A  /  x ]_ { y  |  ph }  <->  z  e.  { y  |  [. A  /  x ]. ph }
) )
1211eqrdv 2387 1  |-  ( A  e.  V  ->  [_ A  /  x ]_ { y  |  ph }  =  { y  |  [. A  /  x ]. ph }
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1649   [wsb 1655    e. wcel 1717   {cab 2375   [.wsbc 3106   [_csb 3196
This theorem is referenced by:  csbsng  3812  csbunig  3967  csbxpg  4847  csbrng  5056  csbfv12gALT  5681  abfmpeld  23910  abfmpel  23911  csbdmg  27652  csbingVD  28339  csbsngVD  28348  csbxpgVD  28349  csbrngVD  28351  csbunigVD  28353  csbfv12gALTVD  28354
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-v 2903  df-sbc 3107  df-csb 3197
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