MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  csbsng Structured version   Visualization version   Unicode version

Theorem csbsng 4021
Description: Distribute proper substitution through the singleton of a class. csbsng 4021 is derived from the virtual deduction proof csbsngVD 37353. (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 3801 . . 3  |-  [_ A  /  x ]_ { y  |  y  =  B }  =  { y  |  [. A  /  x ]. y  =  B }
2 sbceq2g 3783 . . . 4  |-  ( A  e.  V  ->  ( [. A  /  x ]. y  =  B  <->  y  =  [_ A  /  x ]_ B ) )
32abbidv 2589 . . 3  |-  ( A  e.  V  ->  { y  |  [. A  /  x ]. y  =  B }  =  { y  |  y  =  [_ A  /  x ]_ B } )
41, 3syl5eq 2517 . 2  |-  ( A  e.  V  ->  [_ A  /  x ]_ { y  |  y  =  B }  =  { y  |  y  =  [_ A  /  x ]_ B } )
5 df-sn 3960 . . 3  |-  { B }  =  { y  |  y  =  B }
65csbeq2i 3786 . 2  |-  [_ A  /  x ]_ { B }  =  [_ A  /  x ]_ { y  |  y  =  B }
7 df-sn 3960 . 2  |-  { [_ A  /  x ]_ B }  =  { y  |  y  =  [_ A  /  x ]_ B }
84, 6, 73eqtr4g 2530 1  |-  ( A  e.  V  ->  [_ A  /  x ]_ { B }  =  { [_ A  /  x ]_ B }
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1452    e. wcel 1904   {cab 2457   [.wsbc 3255   [_csb 3349   {csn 3959
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-tru 1455  df-fal 1458  df-ex 1672  df-nf 1676  df-sb 1806  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-in 3397  df-ss 3404  df-nul 3723  df-sn 3960
This theorem is referenced by:  csbprg  4022  csbopg2  31795  csbpredg  31797  csbfv12gALTOLD  37276  csbfv12gALTVD  37359
  Copyright terms: Public domain W3C validator