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Theorem csbie2 2977
Description: Conversion of implicit substitution to explicit substitution into a class. (Contributed by NM, 27-Aug-2007.)
Hypotheses
Ref Expression
csbie2t.1  |-  A  e. 
_V
csbie2t.2  |-  B  e. 
_V
csbie2.3  |-  ( ( x  =  A  /\  y  =  B )  ->  C  =  D )
Assertion
Ref Expression
csbie2  |-  [_ A  /  x ]_ [_ B  /  y ]_ C  =  D
Distinct variable groups:    x, y, A   
x, B, y    x, D, y
Allowed substitution hints:    C( x, y)

Proof of Theorem csbie2
StepHypRef Expression
1 csbie2.3 . . 3  |-  ( ( x  =  A  /\  y  =  B )  ->  C  =  D )
21gen2 1384 . 2  |-  A. x A. y ( ( x  =  A  /\  y  =  B )  ->  C  =  D )
3 csbie2t.1 . . 3  |-  A  e. 
_V
4 csbie2t.2 . . 3  |-  B  e. 
_V
53, 4csbie2t 2976 . 2  |-  ( A. x A. y ( ( x  =  A  /\  y  =  B )  ->  C  =  D )  ->  [_ A  /  x ]_ [_ B  /  y ]_ C  =  D
)
62, 5ax-mp 7 1  |-  [_ A  /  x ]_ [_ B  /  y ]_ C  =  D
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102   A.wal 1287    = wceq 1289    e. wcel 1438   _Vcvv 2619   [_csb 2933
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-sbc 2841  df-csb 2934
This theorem is referenced by:  fsumcnv  10831
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