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Theorem csbabgOLD 36899
Description: Move substitution into a class abstraction. (Contributed by NM, 13-Dec-2005.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) Obsolete as of 19-Aug-2018. Use csbab 3822 instead. (New usage is discouraged.) (Proof modification is discouraged.)
Assertion
Ref Expression
csbabgOLD  |-  ( 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 csbabgOLD
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 sbccom 3368 . . . 4  |-  ( [. z  /  y ]. [. A  /  x ]. ph  <->  [. A  /  x ]. [. z  / 
y ]. ph )
2 df-clab 2406 . . . . 5  |-  ( z  e.  { y  | 
[. A  /  x ]. ph }  <->  [ z  /  y ] [. A  /  x ]. ph )
3 sbsbc 3300 . . . . 5  |-  ( [ z  /  y ]
[. A  /  x ]. ph  <->  [. z  /  y ]. [. A  /  x ]. ph )
42, 3bitri 252 . . . 4  |-  ( z  e.  { y  | 
[. A  /  x ]. ph }  <->  [. z  / 
y ]. [. A  /  x ]. ph )
5 df-clab 2406 . . . . . 6  |-  ( z  e.  { y  | 
ph }  <->  [ z  /  y ] ph )
6 sbsbc 3300 . . . . . 6  |-  ( [ z  /  y ]
ph 
<-> 
[. z  /  y ]. ph )
75, 6bitri 252 . . . . 5  |-  ( z  e.  { y  | 
ph }  <->  [. z  / 
y ]. ph )
87sbcbii 3352 . . . 4  |-  ( [. A  /  x ]. z  e.  { y  |  ph } 
<-> 
[. A  /  x ]. [. z  /  y ]. ph )
91, 4, 83bitr4i 280 . . 3  |-  ( z  e.  { y  | 
[. A  /  x ]. ph }  <->  [. A  /  x ]. z  e.  {
y  |  ph }
)
10 sbcel2gOLD 36591 . . 3  |-  ( A  e.  V  ->  ( [. A  /  x ]. z  e.  { y  |  ph }  <->  z  e.  [_ A  /  x ]_ { y  |  ph } ) )
119, 10syl5rbb 261 . 2  |-  ( A  e.  V  ->  (
z  e.  [_ A  /  x ]_ { y  |  ph }  <->  z  e.  { y  |  [. A  /  x ]. ph }
) )
1211eqrdv 2417 1  |-  ( A  e.  V  ->  [_ A  /  x ]_ { y  |  ph }  =  { y  |  [. A  /  x ]. ph }
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1437   [wsb 1786    e. wcel 1867   {cab 2405   [.wsbc 3296   [_csb 3392
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 1838  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-v 3080  df-sbc 3297  df-csb 3393
This theorem is referenced by:  csbunigOLD  36900  csbfv12gALTOLD  36901  csbxpgOLD  36902  csbrngOLD  36905  csbingVD  36969  csbsngVD  36978  csbxpgVD  36979  csbrngVD  36981  csbunigVD  36983  csbfv12gALTVD  36984
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