Theorem csbnegg 8305

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 6009	. 2 |- ( A e. V -> [_ A / x ]_ ( 0 - B ) = ( 0 - [_ A / x ]_ B ) )
2		df-neg 8281	. . 3 |- -u B = ( 0 - B )
3	2	csbeq2i 3128	. 2 |- [_ A / x ]_ -u B = [_ A / x ]_ ( 0 - B )
4		df-neg 8281	. 2 |- -u [_ A / x ]_ B = ( 0 - [_ A / x ]_ B )
5	1, 3, 4	3eqtr4g 2265	1 |- ( A e. V -> [_ A / x ]_ -u B = -u [_ A / x ]_ B )

Syntax hints:   -> wi 4   = wceq 1373   e. wcel 2178   [_csb 3101  (class class class)co 5967   0cc0 7960   - cmin 8278   -ucneg 8279

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-rex 2492  df-v 2778  df-sbc 3006  df-csb 3102  df-un 3178  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-br 4060  df-iota 5251  df-fv 5298  df-ov 5970  df-neg 8281

This theorem is referenced by: (None)