Theorem csbnegg 5376

Description: Move class substitution in and out of the negative of a number.

Assertion

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

Proof of Theorem csbnegg

Step	Hyp	Ref	Expression
1		ax-17 973	. . . 4 |- (A e. C -> A.y A e. C)
2		ax-17 973	. . . . 5 |- (z e. A -> A.y z e. A)
3	2	hbcsb1g 2027	. . . 4 |- (A e. C -> (z e. [_A / y]_[_y / x]_B -> A.y z e. [_A / y]_[_y / x]_B))
4	1, 3	hbnegd 5375	. . 3 |- (A e. C -> (z e. -u[_A / y]_[_y / x]_B -> A.y z e. -u[_A / y]_[_y / x]_B))
5		csbeq1a 2009	. . . . 5 |- (y = A -> [_y / x]_B = [_A / y]_[_y / x]_B)
6	5	negeqd 5373	. . . 4 |- (y = A -> -u[_y / x]_B = -u[_A / y]_[_y / x]_B)
7		a9e 1127	. . . . 5 |- E.x x = y
8		visset 1816	. . . . . . . 8 |- y e. V
9		ax-17 973	. . . . . . . 8 |- (z e. y -> A.x z e. y)
10	8, 9	hbcsb1 2028	. . . . . . 7 |- (z e. [_y / x]_-uB -> A.x z e. [_y / x]_-uB)
11	8, 9	hbcsb1 2028	. . . . . . . 8 |- (z e. [_y / x]_B -> A.x z e. [_y / x]_B)
12	11	hbneg 5374	. . . . . . 7 |- (z e. -u[_y / x]_B -> A.x z e. -u[_y / x]_B)
13	10, 12	hbeq 1568	. . . . . 6 |- ([_y / x]_-uB = -u[_y / x]_B -> A.x[_y / x]_-uB = -u[_y / x]_B)
14		csbeq1a 2009	. . . . . . 7 |- (x = y -> -uB = [_y / x]_-uB)
15		csbeq1a 2009	. . . . . . . 8 |- (x = y -> B = [_y / x]_B)
16	15	negeqd 5373	. . . . . . 7 |- (x = y -> -uB = -u[_y / x]_B)
17	14, 16	eqtr3d 1512	. . . . . 6 |- (x = y -> [_y / x]_-uB = -u[_y / x]_B)
18	13, 17	19.23ai 1066	. . . . 5 |- (E.x x = y -> [_y / x]_-uB = -u[_y / x]_B)
19	7, 18	ax-mp 7	. . . 4 |- [_y / x]_-uB = -u[_y / x]_B
20	6, 19	syl5eq 1522	. . 3 |- (y = A -> [_y / x]_-uB = -u[_A / y]_[_y / x]_B)
21	4, 20	csbiegf 2034	. 2 |- (A e. C -> [_A / y]_[_y / x]_-uB = -u[_A / y]_[_y / x]_B)
22		csbcog 2010	. 2 |- (A e. C -> [_A / y]_[_y / x]_-uB = [_A / x]_-uB)
23		csbcog 2010	. . 3 |- (A e. C -> [_A / y]_[_y / x]_B = [_A / x]_B)
24	23	negeqd 5373	. 2 |- (A e. C -> -u[_A / y]_[_y / x]_B = -u[_A / x]_B)
25	21, 22, 24	3eqtr3d 1518	1 |- (A e. C -> [_A / x]_-uB = -u[_A / x]_B)

Syntax hints:   -> wi 3   = wceq 958   e. wcel 960  E.wex 982  [_csb 2004  -ucneg 5305

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-9 967  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708  ax-pow 2748  ax-pr 2785

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-rex 1653  df-v 1815  df-sbc 1945  df-csb 2005  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420  df-uni 2508  df-br 2625  df-opab 2672  df-xp 3190  df-cnv 3192  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fv 3204  df-opr 3971  df-neg 5370

Copyright terms: Public domain