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Theorem csbnegg 9049
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 5892 . 2  |-  ( A  e.  V  ->  [_ A  /  x ]_ ( 0  -  B )  =  ( 0  -  [_ A  /  x ]_ B
) )
2 df-neg 9040 . . 3  |-  -u B  =  ( 0  -  B )
32csbeq2i 3107 . 2  |-  [_ A  /  x ]_ -u B  =  [_ A  /  x ]_ ( 0  -  B
)
4 df-neg 9040 . 2  |-  -u [_ A  /  x ]_ B  =  ( 0  -  [_ A  /  x ]_ B
)
51, 3, 43eqtr4g 2340 1  |-  ( A  e.  V  ->  [_ A  /  x ]_ -u B  =  -u [_ A  /  x ]_ B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    e. wcel 1684   [_csb 3081  (class class class)co 5858   0cc0 8737    - cmin 9037   -ucneg 9038
This theorem is referenced by:  dvfsum2  19381
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-iota 5219  df-fv 5263  df-ov 5861  df-neg 9040
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