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Theorem csbnegg 7883
Description: Move class substitution in and out of the negative of a number. (Contributed by NM, 1-Mar-2008.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
Assertion
Ref Expression
csbnegg  |-  ( A  e.  V  ->  [_ A  /  x ]_ -u B  =  -u [_ A  /  x ]_ B )

Proof of Theorem csbnegg
StepHypRef Expression
1 csbov2g 5766 . 2  |-  ( A  e.  V  ->  [_ A  /  x ]_ ( 0  -  B )  =  ( 0  -  [_ A  /  x ]_ B
) )
2 df-neg 7859 . . 3  |-  -u B  =  ( 0  -  B )
32csbeq2i 2995 . 2  |-  [_ A  /  x ]_ -u B  =  [_ A  /  x ]_ ( 0  -  B
)
4 df-neg 7859 . 2  |-  -u [_ A  /  x ]_ B  =  ( 0  -  [_ A  /  x ]_ B
)
51, 3, 43eqtr4g 2172 1  |-  ( A  e.  V  ->  [_ A  /  x ]_ -u B  =  -u [_ A  /  x ]_ B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1314    e. wcel 1463   [_csb 2971  (class class class)co 5728   0cc0 7547    - cmin 7856   -ucneg 7857
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-rex 2396  df-v 2659  df-sbc 2879  df-csb 2972  df-un 3041  df-sn 3499  df-pr 3500  df-op 3502  df-uni 3703  df-br 3896  df-iota 5046  df-fv 5089  df-ov 5731  df-neg 7859
This theorem is referenced by: (None)
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