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Theorem csb2 3074
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 3073 . 2  |-  [_ A  /  x ]_ B  =  { y  |  [. A  /  x ]. y  e.  B }
2 sbc5 3001 . . 3  |-  ( [. A  /  x ]. y  e.  B  <->  E. x ( x  =  A  /\  y  e.  B ) )
32abbii 2305 . 2  |-  { y  |  [. A  /  x ]. y  e.  B }  =  { y  |  E. x ( x  =  A  /\  y  e.  B ) }
41, 3eqtri 2210 1  |-  [_ A  /  x ]_ B  =  { y  |  E. x ( x  =  A  /\  y  e.  B ) }
Colors of variables: wff set class
Syntax hints:    /\ wa 104    = wceq 1364   E.wex 1503    e. wcel 2160   {cab 2175   [.wsbc 2977   [_csb 3072
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-v 2754  df-sbc 2978  df-csb 3073
This theorem is referenced by: (None)
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