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Theorem csbnegg 9065
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 5908 . 2  |-  ( A  e.  V  ->  [_ A  /  x ]_ ( 0  -  B )  =  ( 0  -  [_ A  /  x ]_ B
) )
2 df-neg 9056 . . 3  |-  -u B  =  ( 0  -  B )
32csbeq2i 3120 . 2  |-  [_ A  /  x ]_ -u B  =  [_ A  /  x ]_ ( 0  -  B
)
4 df-neg 9056 . 2  |-  -u [_ A  /  x ]_ B  =  ( 0  -  [_ A  /  x ]_ B
)
51, 3, 43eqtr4g 2353 1  |-  ( A  e.  V  ->  [_ A  /  x ]_ -u B  =  -u [_ A  /  x ]_ B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1632    e. wcel 1696   [_csb 3094  (class class class)co 5874   0cc0 8753    - cmin 9053   -ucneg 9054
This theorem is referenced by:  dvfsum2  19397
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-iota 5235  df-fv 5279  df-ov 5877  df-neg 9056
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