ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  csbsng Unicode version

Theorem csbsng 3644
Description: Distribute proper substitution through the singleton of a class. (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 3110 . . 3  |-  ( A  e.  V  ->  [_ A  /  x ]_ { y  |  y  =  B }  =  { y  |  [. A  /  x ]. y  =  B } )
2 sbceq2g 3071 . . . 4  |-  ( A  e.  V  ->  ( [. A  /  x ]. y  =  B  <->  y  =  [_ A  /  x ]_ B ) )
32abbidv 2288 . . 3  |-  ( A  e.  V  ->  { y  |  [. A  /  x ]. y  =  B }  =  { y  |  y  =  [_ A  /  x ]_ B } )
41, 3eqtrd 2203 . 2  |-  ( A  e.  V  ->  [_ A  /  x ]_ { y  |  y  =  B }  =  { y  |  y  =  [_ A  /  x ]_ B } )
5 df-sn 3589 . . 3  |-  { B }  =  { y  |  y  =  B }
65csbeq2i 3076 . 2  |-  [_ A  /  x ]_ { B }  =  [_ A  /  x ]_ { y  |  y  =  B }
7 df-sn 3589 . 2  |-  { [_ A  /  x ]_ B }  =  { y  |  y  =  [_ A  /  x ]_ B }
84, 6, 73eqtr4g 2228 1  |-  ( A  e.  V  ->  [_ A  /  x ]_ { B }  =  { [_ A  /  x ]_ B }
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1348    e. wcel 2141   {cab 2156   [.wsbc 2955   [_csb 3049   {csn 3583
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-sbc 2956  df-csb 3050  df-sn 3589
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator