Theorem csbnegg 6520

Description: Move class substitution in and out of the negative of a number. (The proof was shortened by Andrew Salmon, 22-Oct-2011.)

Assertion

Ref	Expression
csbnegg	|- (A e. C -> [_A / x]_-uB = -u[_A / x]_B)

Proof of Theorem csbnegg

Step	Hyp	Ref	Expression
1		csbopr2g 4914	. 2 |- (A e. C -> [_A / x]_(0 - B) = (0 - [_A / x]_B))
2		df-neg 6513	. . 3 |- -uB = (0 - B)
3	2	csbeq2i 2563	. 2 |- (A e. C -> [_A / x]_-uB = [_A / x]_(0 - B))
4		df-neg 6513	. . 3 |- -u[_A / x]_B = (0 - [_A / x]_B)
5	4	a1i 8	. 2 |- (A e. C -> -u[_A / x]_B = (0 - [_A / x]_B))
6	1, 3, 5	3eqtr4d 1937	1 |- (A e. C -> [_A / x]_-uB = -u[_A / x]_B)

Syntax hints:   -> wi 3   = wceq 1298   e. wcel 1300  [_csb 2540  (class class class)co 4884  0cc0 6386   - cmin 6445  -ucneg 6446

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-5 1302  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-rex 2110  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-xp 4000  df-cnv 4002  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fv 4014  df-opr 4886  df-neg 6513

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