Theorem csbnegg 8067

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 5859	. 2 |- ( A e. V -> [_ A / x ]_ ( 0 - B ) = ( 0 - [_ A / x ]_ B ) )
2		df-neg 8043	. . 3 |- -u B = ( 0 - B )
3	2	csbeq2i 3058	. 2 |- [_ A / x ]_ -u B = [_ A / x ]_ ( 0 - B )
4		df-neg 8043	. 2 |- -u [_ A / x ]_ B = ( 0 - [_ A / x ]_ B )
5	1, 3, 4	3eqtr4g 2215	1 |- ( A e. V -> [_ A / x ]_ -u B = -u [_ A / x ]_ B )

Syntax hints:   -> wi 4   = wceq 1335   e. wcel 2128   [_csb 3031  (class class class)co 5821   0cc0 7726   - cmin 8040   -ucneg 8041

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 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-rex 2441  df-v 2714  df-sbc 2938  df-csb 3032  df-un 3106  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-br 3966  df-iota 5134  df-fv 5177  df-ov 5824  df-neg 8043

This theorem is referenced by: (None)