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Theorem csb2 2977
Description: Alternate expression for the proper substitution into a class, without referencing substitution into a wff. Note that  x can be free in  B but cannot occur in  A. (Contributed by NM, 2-Dec-2013.)
Assertion
Ref Expression
csb2  |-  [_ A  /  x ]_ B  =  { y  |  E. x ( x  =  A  /\  y  e.  B ) }
Distinct variable groups:    x, y, A   
y, B
Allowed substitution hint:    B( x)

Proof of Theorem csb2
StepHypRef Expression
1 df-csb 2976 . 2  |-  [_ A  /  x ]_ B  =  { y  |  [. A  /  x ]. y  e.  B }
2 sbc5 2905 . . 3  |-  ( [. A  /  x ]. y  e.  B  <->  E. x ( x  =  A  /\  y  e.  B ) )
32abbii 2233 . 2  |-  { y  |  [. A  /  x ]. y  e.  B }  =  { y  |  E. x ( x  =  A  /\  y  e.  B ) }
41, 3eqtri 2138 1  |-  [_ A  /  x ]_ B  =  { y  |  E. x ( x  =  A  /\  y  e.  B ) }
Colors of variables: wff set class
Syntax hints:    /\ wa 103    = wceq 1316   E.wex 1453    e. wcel 1465   {cab 2103   [.wsbc 2882   [_csb 2975
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 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-v 2662  df-sbc 2883  df-csb 2976
This theorem is referenced by: (None)
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