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Theorem csbnegg 7928
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 5780 . 2  |-  ( A  e.  V  ->  [_ A  /  x ]_ ( 0  -  B )  =  ( 0  -  [_ A  /  x ]_ B
) )
2 df-neg 7904 . . 3  |-  -u B  =  ( 0  -  B )
32csbeq2i 2999 . 2  |-  [_ A  /  x ]_ -u B  =  [_ A  /  x ]_ ( 0  -  B
)
4 df-neg 7904 . 2  |-  -u [_ A  /  x ]_ B  =  ( 0  -  [_ A  /  x ]_ B
)
51, 3, 43eqtr4g 2175 1  |-  ( A  e.  V  ->  [_ A  /  x ]_ -u B  =  -u [_ A  /  x ]_ B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1316    e. wcel 1465   [_csb 2975  (class class class)co 5742   0cc0 7588    - cmin 7901   -ucneg 7902
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-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-rex 2399  df-v 2662  df-sbc 2883  df-csb 2976  df-un 3045  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-br 3900  df-iota 5058  df-fv 5101  df-ov 5745  df-neg 7904
This theorem is referenced by: (None)
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