Theorem csbnegg 8473

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

Step	Hyp	Ref	Expression
1		csbov2g 6094	. 2 |- ( A e. V -> [_ A / x ]_ ( 0 - B ) = ( 0 - [_ A / x ]_ B ) )
2		df-neg 8449	. . 3 |- -u B = ( 0 - B )
3	2	csbeq2i 3167	. 2 |- [_ A / x ]_ -u B = [_ A / x ]_ ( 0 - B )
4		df-neg 8449	. 2 |- -u [_ A / x ]_ B = ( 0 - [_ A / x ]_ B )
5	1, 3, 4	3eqtr4g 2292	1 |- ( A e. V -> [_ A / x ]_ -u B = -u [_ A / x ]_ B )

Syntax hints:   -> wi 4   = wceq 1398   e. wcel 2205   [_csb 3140  (class class class)co 6052   0cc0 8129   - cmin 8446   -ucneg 8447

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2216

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-rex 2528  df-v 2817  df-sbc 3045  df-csb 3141  df-un 3217  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-br 4112  df-iota 5314  df-fv 5362  df-ov 6055  df-neg 8449

This theorem is referenced by: (None)