Theorem csbnegg 8224

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 5963	. 2 |- ( A e. V -> [_ A / x ]_ ( 0 - B ) = ( 0 - [_ A / x ]_ B ) )
2		df-neg 8200	. . 3 |- -u B = ( 0 - B )
3	2	csbeq2i 3111	. 2 |- [_ A / x ]_ -u B = [_ A / x ]_ ( 0 - B )
4		df-neg 8200	. 2 |- -u [_ A / x ]_ B = ( 0 - [_ A / x ]_ B )
5	1, 3, 4	3eqtr4g 2254	1 |- ( A e. V -> [_ A / x ]_ -u B = -u [_ A / x ]_ B )

Syntax hints:   -> wi 4   = wceq 1364   e. wcel 2167   [_csb 3084  (class class class)co 5922   0cc0 7879   - cmin 8197   -ucneg 8198

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-rex 2481  df-v 2765  df-sbc 2990  df-csb 3085  df-un 3161  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-br 4034  df-iota 5219  df-fv 5266  df-ov 5925  df-neg 8200

This theorem is referenced by: (None)