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Theorem csbsng 4030
Description: Distribute proper substitution through the singleton of a class. csbsng 4030 is derived from the virtual deduction proof csbsngVD 37290. (Contributed by Alan Sare, 10-Nov-2012.)
Assertion
Ref Expression
csbsng  |-  ( A  e.  V  ->  [_ A  /  x ]_ { B }  =  { [_ A  /  x ]_ B }
)

Proof of Theorem csbsng
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 csbab 3797 . . 3  |-  [_ A  /  x ]_ { y  |  y  =  B }  =  { y  |  [. A  /  x ]. y  =  B }
2 sbceq2g 3779 . . . 4  |-  ( A  e.  V  ->  ( [. A  /  x ]. y  =  B  <->  y  =  [_ A  /  x ]_ B ) )
32abbidv 2569 . . 3  |-  ( A  e.  V  ->  { y  |  [. A  /  x ]. y  =  B }  =  { y  |  y  =  [_ A  /  x ]_ B } )
41, 3syl5eq 2497 . 2  |-  ( A  e.  V  ->  [_ A  /  x ]_ { y  |  y  =  B }  =  { y  |  y  =  [_ A  /  x ]_ B } )
5 df-sn 3969 . . 3  |-  { B }  =  { y  |  y  =  B }
65csbeq2i 3782 . 2  |-  [_ A  /  x ]_ { B }  =  [_ A  /  x ]_ { y  |  y  =  B }
7 df-sn 3969 . 2  |-  { [_ A  /  x ]_ B }  =  { y  |  y  =  [_ A  /  x ]_ B }
84, 6, 73eqtr4g 2510 1  |-  ( A  e.  V  ->  [_ A  /  x ]_ { B }  =  { [_ A  /  x ]_ B }
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1444    e. wcel 1887   {cab 2437   [.wsbc 3267   [_csb 3363   {csn 3968
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-tru 1447  df-fal 1450  df-ex 1664  df-nf 1668  df-sb 1798  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-in 3411  df-ss 3418  df-nul 3732  df-sn 3969
This theorem is referenced by:  csbprg  4031  csbopg2  31725  csbpredg  31727  csbfv12gALTOLD  37213  csbfv12gALTVD  37296
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