Theorem csbnegg 5344

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

Assertion

Ref	Expression
csbnegg	⊢ (A ∈ C → [A / x]-B = -[A / x]B)

Proof of Theorem csbnegg

Step	Hyp	Ref	Expression
1		ax-17 969	. . . 4 ⊢ (A ∈ C → ∀y A ∈ C)
2		ax-17 969	. . . . 5 ⊢ (z ∈ A → ∀y z ∈ A)
3	2	hbcsb1g 2020	. . . 4 ⊢ (A ∈ C → (z ∈ [A / y][y / x]B → ∀y z ∈ [A / y][y / x]B))
4	1, 3	hbnegd 5343	. . 3 ⊢ (A ∈ C → (z ∈ -[A / y][y / x]B → ∀y z ∈ -[A / y][y / x]B))
5		csbeq1a 2002	. . . . 5 ⊢ (y = A → [y / x]B = [A / y][y / x]B)
6	5	negeqd 5341	. . . 4 ⊢ (y = A → -[y / x]B = -[A / y][y / x]B)
7		a9e 1123	. . . . 5 ⊢ ∃x x = y
8		visset 1809	. . . . . . . 8 ⊢ y ∈ V
9		ax-17 969	. . . . . . . 8 ⊢ (z ∈ y → ∀x z ∈ y)
10	8, 9	hbcsb1 2021	. . . . . . 7 ⊢ (z ∈ [y / x]-B → ∀x z ∈ [y / x]-B)
11	8, 9	hbcsb1 2021	. . . . . . . 8 ⊢ (z ∈ [y / x]B → ∀x z ∈ [y / x]B)
12	11	hbneg 5342	. . . . . . 7 ⊢ (z ∈ -[y / x]B → ∀x z ∈ -[y / x]B)
13	10, 12	hbeq 1562	. . . . . 6 ⊢ ([y / x]-B = -[y / x]B → ∀x[y / x]-B = -[y / x]B)
14		csbeq1a 2002	. . . . . . 7 ⊢ (x = y → -B = [y / x]-B)
15		csbeq1a 2002	. . . . . . . 8 ⊢ (x = y → B = [y / x]B)
16	15	negeqd 5341	. . . . . . 7 ⊢ (x = y → -B = -[y / x]B)
17	14, 16	eqtr3d 1506	. . . . . 6 ⊢ (x = y → [y / x]-B = -[y / x]B)
18	13, 17	19.23ai 1062	. . . . 5 ⊢ (∃x x = y → [y / x]-B = -[y / x]B)
19	7, 18	ax-mp 7	. . . 4 ⊢ [y / x]-B = -[y / x]B
20	6, 19	syl5eq 1516	. . 3 ⊢ (y = A → [y / x]-B = -[A / y][y / x]B)
21	4, 20	csbiegf 2027	. 2 ⊢ (A ∈ C → [A / y][y / x]-B = -[A / y][y / x]B)
22		csbcog 2003	. 2 ⊢ (A ∈ C → [A / y][y / x]-B = [A / x]-B)
23		csbcog 2003	. . 3 ⊢ (A ∈ C → [A / y][y / x]B = [A / x]B)
24	23	negeqd 5341	. 2 ⊢ (A ∈ C → -[A / y][y / x]B = -[A / x]B)
25	21, 22, 24	3eqtr3d 1512	1 ⊢ (A ∈ C → [A / x]-B = -[A / x]B)

Syntax hints:   → wi 3   = wceq 954   ∈ wcel 956  ∃wex 978  [csb 1997  -cneg 5273

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-9 963  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-sep 2698  ax-pow 2737  ax-pr 2774

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 776  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-rex 1647  df-v 1808  df-sbc 1938  df-csb 1998  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-nul 2277  df-pw 2398  df-sn 2408  df-pr 2409  df-op 2412  df-uni 2499  df-br 2615  df-opab 2662  df-xp 3179  df-cnv 3181  df-dm 3183  df-rn 3184  df-res 3185  df-ima 3186  df-fv 3193  df-opr 3956  df-neg 5338

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