Theorem csbdif 4458

Description: Distribution of class substitution over difference of two classes. (Contributed by ML, 14-Jul-2020.)

Assertion

Ref	Expression
csbdif	⊢ ⦋𝐴 / 𝑥⦌(𝐵 ∖ 𝐶) = (⦋𝐴 / 𝑥⦌𝐵 ∖ ⦋𝐴 / 𝑥⦌𝐶)

Proof of Theorem csbdif

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		csbeq1 3835	. . . 4 ⊢ (𝑦 = 𝐴 → ⦋𝑦 / 𝑥⦌(𝐵 ∖ 𝐶) = ⦋𝐴 / 𝑥⦌(𝐵 ∖ 𝐶))
2		csbeq1 3835	. . . . 5 ⊢ (𝑦 = 𝐴 → ⦋𝑦 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐵)
3		csbeq1 3835	. . . . 5 ⊢ (𝑦 = 𝐴 → ⦋𝑦 / 𝑥⦌𝐶 = ⦋𝐴 / 𝑥⦌𝐶)
4	2, 3	difeq12d 4058	. . . 4 ⊢ (𝑦 = 𝐴 → (⦋𝑦 / 𝑥⦌𝐵 ∖ ⦋𝑦 / 𝑥⦌𝐶) = (⦋𝐴 / 𝑥⦌𝐵 ∖ ⦋𝐴 / 𝑥⦌𝐶))
5	1, 4	eqeq12d 2754	. . 3 ⊢ (𝑦 = 𝐴 → (⦋𝑦 / 𝑥⦌(𝐵 ∖ 𝐶) = (⦋𝑦 / 𝑥⦌𝐵 ∖ ⦋𝑦 / 𝑥⦌𝐶) ↔ ⦋𝐴 / 𝑥⦌(𝐵 ∖ 𝐶) = (⦋𝐴 / 𝑥⦌𝐵 ∖ ⦋𝐴 / 𝑥⦌𝐶)))
6		vex 3436	. . . 4 ⊢ 𝑦 ∈ V
7		nfcsb1v 3857	. . . . 5 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐵
8		nfcsb1v 3857	. . . . 5 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐶
9	7, 8	nfdif 4060	. . . 4 ⊢ Ⅎ𝑥(⦋𝑦 / 𝑥⦌𝐵 ∖ ⦋𝑦 / 𝑥⦌𝐶)
10		csbeq1a 3846	. . . . 5 ⊢ (𝑥 = 𝑦 → 𝐵 = ⦋𝑦 / 𝑥⦌𝐵)
11		csbeq1a 3846	. . . . 5 ⊢ (𝑥 = 𝑦 → 𝐶 = ⦋𝑦 / 𝑥⦌𝐶)
12	10, 11	difeq12d 4058	. . . 4 ⊢ (𝑥 = 𝑦 → (𝐵 ∖ 𝐶) = (⦋𝑦 / 𝑥⦌𝐵 ∖ ⦋𝑦 / 𝑥⦌𝐶))
13	6, 9, 12	csbief 3867	. . 3 ⊢ ⦋𝑦 / 𝑥⦌(𝐵 ∖ 𝐶) = (⦋𝑦 / 𝑥⦌𝐵 ∖ ⦋𝑦 / 𝑥⦌𝐶)
14	5, 13	vtoclg 3505	. 2 ⊢ (𝐴 ∈ V → ⦋𝐴 / 𝑥⦌(𝐵 ∖ 𝐶) = (⦋𝐴 / 𝑥⦌𝐵 ∖ ⦋𝐴 / 𝑥⦌𝐶))
15		dif0 4306	. . . 4 ⊢ (∅ ∖ ∅) = ∅
16	15	a1i 11	. . 3 ⊢ (¬ 𝐴 ∈ V → (∅ ∖ ∅) = ∅)
17		csbprc 4340	. . . 4 ⊢ (¬ 𝐴 ∈ V → ⦋𝐴 / 𝑥⦌𝐵 = ∅)
18		csbprc 4340	. . . 4 ⊢ (¬ 𝐴 ∈ V → ⦋𝐴 / 𝑥⦌𝐶 = ∅)
19	17, 18	difeq12d 4058	. . 3 ⊢ (¬ 𝐴 ∈ V → (⦋𝐴 / 𝑥⦌𝐵 ∖ ⦋𝐴 / 𝑥⦌𝐶) = (∅ ∖ ∅))
20		csbprc 4340	. . 3 ⊢ (¬ 𝐴 ∈ V → ⦋𝐴 / 𝑥⦌(𝐵 ∖ 𝐶) = ∅)
21	16, 19, 20	3eqtr4rd 2789	. 2 ⊢ (¬ 𝐴 ∈ V → ⦋𝐴 / 𝑥⦌(𝐵 ∖ 𝐶) = (⦋𝐴 / 𝑥⦌𝐵 ∖ ⦋𝐴 / 𝑥⦌𝐶))
22	14, 21	pm2.61i 182	1 ⊢ ⦋𝐴 / 𝑥⦌(𝐵 ∖ 𝐶) = (⦋𝐴 / 𝑥⦌𝐵 ∖ ⦋𝐴 / 𝑥⦌𝐶)

Syntax hints:  ¬ wn 3   = wceq 1539   ∈ wcel 2106  Vcvv 3432  ⦋csb 3832   ∖ cdif 3884  ∅c0 4256

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-nul 4257

This theorem is referenced by: (None)