Theorem csbnegg 7209

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 5546	. 2 |- ( A e. V -> [_ A / x ]_ ( 0 - B ) = ( 0 - [_ A / x ]_ B ) )
2		df-neg 7185	. . 3 |- -u B = ( 0 - B )
3	2	csbeq2i 2876	. 2 |- [_ A / x ]_ -u B = [_ A / x ]_ ( 0 - B )
4		df-neg 7185	. 2 |- -u [_ A / x ]_ B = ( 0 - [_ A / x ]_ B )
5	1, 3, 4	3eqtr4g 2097	1 |- ( A e. V -> [_ A / x ]_ -u B = -u [_ A / x ]_ B )

Syntax hints:   -> wi 4   = wceq 1243   e. wcel 1393   [_csb 2852  (class class class)co 5512   0cc0 6889   - cmin 7182   -ucneg 7183

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-rex 2312  df-v 2559  df-sbc 2765  df-csb 2853  df-un 2922  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-iota 4867  df-fv 4910  df-ov 5515  df-neg 7185

This theorem is referenced by: (None)