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Theorem csbsng 3867
Description: Distribute proper substitution through the singleton of a class. csbsng 3867 is derived from the virtual deduction proof csbsngVD 29005. (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 csbabg 3310 . . 3  |-  ( A  e.  V  ->  [_ A  /  x ]_ { y  |  y  =  B }  =  { y  |  [. A  /  x ]. y  =  B } )
2 sbceq2g 3273 . . . 4  |-  ( A  e.  V  ->  ( [. A  /  x ]. y  =  B  <->  y  =  [_ A  /  x ]_ B ) )
32abbidv 2550 . . 3  |-  ( A  e.  V  ->  { y  |  [. A  /  x ]. y  =  B }  =  { y  |  y  =  [_ A  /  x ]_ B } )
41, 3eqtrd 2468 . 2  |-  ( A  e.  V  ->  [_ A  /  x ]_ { y  |  y  =  B }  =  { y  |  y  =  [_ A  /  x ]_ B } )
5 df-sn 3820 . . 3  |-  { B }  =  { y  |  y  =  B }
65csbeq2i 3277 . 2  |-  [_ A  /  x ]_ { B }  =  [_ A  /  x ]_ { y  |  y  =  B }
7 df-sn 3820 . 2  |-  { [_ A  /  x ]_ B }  =  { y  |  y  =  [_ A  /  x ]_ B }
84, 6, 73eqtr4g 2493 1  |-  ( A  e.  V  ->  [_ A  /  x ]_ { B }  =  { [_ A  /  x ]_ B }
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1652    e. wcel 1725   {cab 2422   [.wsbc 3161   [_csb 3251   {csn 3814
This theorem is referenced by:  csbfv12gALT  5739  csbfv12gALTVD  29011
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-v 2958  df-sbc 3162  df-csb 3252  df-sn 3820
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