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Theorem csbnegg 9235
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 6054 . 2  |-  ( A  e.  V  ->  [_ A  /  x ]_ ( 0  -  B )  =  ( 0  -  [_ A  /  x ]_ B
) )
2 df-neg 9226 . . 3  |-  -u B  =  ( 0  -  B )
32csbeq2i 3220 . 2  |-  [_ A  /  x ]_ -u B  =  [_ A  /  x ]_ ( 0  -  B
)
4 df-neg 9226 . 2  |-  -u [_ A  /  x ]_ B  =  ( 0  -  [_ A  /  x ]_ B
)
51, 3, 43eqtr4g 2444 1  |-  ( A  e.  V  ->  [_ A  /  x ]_ -u B  =  -u [_ A  /  x ]_ B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1649    e. wcel 1717   [_csb 3194  (class class class)co 6020   0cc0 8923    - cmin 9223   -ucneg 9224
This theorem is referenced by:  dvfsum2  19785
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ral 2654  df-rex 2655  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-br 4154  df-iota 5358  df-fv 5402  df-ov 6023  df-neg 9226
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