Theorem csbdif 4504

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 3882	. . . 4 ⊢ (𝑦 = 𝐴 → ⦋𝑦 / 𝑥⦌(𝐵 ∖ 𝐶) = ⦋𝐴 / 𝑥⦌(𝐵 ∖ 𝐶))
2		csbeq1 3882	. . . . 5 ⊢ (𝑦 = 𝐴 → ⦋𝑦 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐵)
3		csbeq1 3882	. . . . 5 ⊢ (𝑦 = 𝐴 → ⦋𝑦 / 𝑥⦌𝐶 = ⦋𝐴 / 𝑥⦌𝐶)
4	2, 3	difeq12d 4107	. . . 4 ⊢ (𝑦 = 𝐴 → (⦋𝑦 / 𝑥⦌𝐵 ∖ ⦋𝑦 / 𝑥⦌𝐶) = (⦋𝐴 / 𝑥⦌𝐵 ∖ ⦋𝐴 / 𝑥⦌𝐶))
5	1, 4	eqeq12d 2752	. . 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 3468	. . . 4 ⊢ 𝑦 ∈ V
7		nfcsb1v 3903	. . . . 5 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐵
8		nfcsb1v 3903	. . . . 5 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐶
9	7, 8	nfdif 4109	. . . 4 ⊢ Ⅎ𝑥(⦋𝑦 / 𝑥⦌𝐵 ∖ ⦋𝑦 / 𝑥⦌𝐶)
10		csbeq1a 3893	. . . . 5 ⊢ (𝑥 = 𝑦 → 𝐵 = ⦋𝑦 / 𝑥⦌𝐵)
11		csbeq1a 3893	. . . . 5 ⊢ (𝑥 = 𝑦 → 𝐶 = ⦋𝑦 / 𝑥⦌𝐶)
12	10, 11	difeq12d 4107	. . . 4 ⊢ (𝑥 = 𝑦 → (𝐵 ∖ 𝐶) = (⦋𝑦 / 𝑥⦌𝐵 ∖ ⦋𝑦 / 𝑥⦌𝐶))
13	6, 9, 12	csbief 3913	. . 3 ⊢ ⦋𝑦 / 𝑥⦌(𝐵 ∖ 𝐶) = (⦋𝑦 / 𝑥⦌𝐵 ∖ ⦋𝑦 / 𝑥⦌𝐶)
14	5, 13	vtoclg 3538	. 2 ⊢ (𝐴 ∈ V → ⦋𝐴 / 𝑥⦌(𝐵 ∖ 𝐶) = (⦋𝐴 / 𝑥⦌𝐵 ∖ ⦋𝐴 / 𝑥⦌𝐶))
15		dif0 4358	. . . 4 ⊢ (∅ ∖ ∅) = ∅
16	15	a1i 11	. . 3 ⊢ (¬ 𝐴 ∈ V → (∅ ∖ ∅) = ∅)
17		csbprc 4389	. . . 4 ⊢ (¬ 𝐴 ∈ V → ⦋𝐴 / 𝑥⦌𝐵 = ∅)
18		csbprc 4389	. . . 4 ⊢ (¬ 𝐴 ∈ V → ⦋𝐴 / 𝑥⦌𝐶 = ∅)
19	17, 18	difeq12d 4107	. . 3 ⊢ (¬ 𝐴 ∈ V → (⦋𝐴 / 𝑥⦌𝐵 ∖ ⦋𝐴 / 𝑥⦌𝐶) = (∅ ∖ ∅))
20		csbprc 4389	. . 3 ⊢ (¬ 𝐴 ∈ V → ⦋𝐴 / 𝑥⦌(𝐵 ∖ 𝐶) = ∅)
21	16, 19, 20	3eqtr4rd 2782	. 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 1540   ∈ wcel 2109  Vcvv 3464  ⦋csb 3879   ∖ cdif 3928  ∅c0 4313

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-rab 3421  df-v 3466  df-sbc 3771  df-csb 3880  df-dif 3934  df-nul 4314

This theorem is referenced by: (None)