Theorem csbun 4389

Description: Distribution of class substitution over union of two classes. (Contributed by Drahflow, 23-Sep-2015.) (Revised by Mario Carneiro, 11-Dec-2016.) (Revised by NM, 13-Sep-2018.)

Assertion

Ref	Expression
csbun	⊢ ⦋𝐴 / 𝑥⦌(𝐵 ∪ 𝐶) = (⦋𝐴 / 𝑥⦌𝐵 ∪ ⦋𝐴 / 𝑥⦌𝐶)

Proof of Theorem csbun

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		csbeq1 3885	. . . 4 ⊢ (𝑦 = 𝐴 → ⦋𝑦 / 𝑥⦌(𝐵 ∪ 𝐶) = ⦋𝐴 / 𝑥⦌(𝐵 ∪ 𝐶))
2		csbeq1 3885	. . . . 5 ⊢ (𝑦 = 𝐴 → ⦋𝑦 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐵)
3		csbeq1 3885	. . . . 5 ⊢ (𝑦 = 𝐴 → ⦋𝑦 / 𝑥⦌𝐶 = ⦋𝐴 / 𝑥⦌𝐶)
4	2, 3	uneq12d 4139	. . . 4 ⊢ (𝑦 = 𝐴 → (⦋𝑦 / 𝑥⦌𝐵 ∪ ⦋𝑦 / 𝑥⦌𝐶) = (⦋𝐴 / 𝑥⦌𝐵 ∪ ⦋𝐴 / 𝑥⦌𝐶))
5	1, 4	eqeq12d 2837	. . 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 3497	. . . 4 ⊢ 𝑦 ∈ V
7		nfcsb1v 3906	. . . . 5 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐵
8		nfcsb1v 3906	. . . . 5 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐶
9	7, 8	nfun 4140	. . . 4 ⊢ Ⅎ𝑥(⦋𝑦 / 𝑥⦌𝐵 ∪ ⦋𝑦 / 𝑥⦌𝐶)
10		csbeq1a 3896	. . . . 5 ⊢ (𝑥 = 𝑦 → 𝐵 = ⦋𝑦 / 𝑥⦌𝐵)
11		csbeq1a 3896	. . . . 5 ⊢ (𝑥 = 𝑦 → 𝐶 = ⦋𝑦 / 𝑥⦌𝐶)
12	10, 11	uneq12d 4139	. . . 4 ⊢ (𝑥 = 𝑦 → (𝐵 ∪ 𝐶) = (⦋𝑦 / 𝑥⦌𝐵 ∪ ⦋𝑦 / 𝑥⦌𝐶))
13	6, 9, 12	csbief 3916	. . 3 ⊢ ⦋𝑦 / 𝑥⦌(𝐵 ∪ 𝐶) = (⦋𝑦 / 𝑥⦌𝐵 ∪ ⦋𝑦 / 𝑥⦌𝐶)
14	5, 13	vtoclg 3567	. 2 ⊢ (𝐴 ∈ V → ⦋𝐴 / 𝑥⦌(𝐵 ∪ 𝐶) = (⦋𝐴 / 𝑥⦌𝐵 ∪ ⦋𝐴 / 𝑥⦌𝐶))
15		un0 4343	. . . 4 ⊢ (∅ ∪ ∅) = ∅
16	15	a1i 11	. . 3 ⊢ (¬ 𝐴 ∈ V → (∅ ∪ ∅) = ∅)
17		csbprc 4357	. . . 4 ⊢ (¬ 𝐴 ∈ V → ⦋𝐴 / 𝑥⦌𝐵 = ∅)
18		csbprc 4357	. . . 4 ⊢ (¬ 𝐴 ∈ V → ⦋𝐴 / 𝑥⦌𝐶 = ∅)
19	17, 18	uneq12d 4139	. . 3 ⊢ (¬ 𝐴 ∈ V → (⦋𝐴 / 𝑥⦌𝐵 ∪ ⦋𝐴 / 𝑥⦌𝐶) = (∅ ∪ ∅))
20		csbprc 4357	. . 3 ⊢ (¬ 𝐴 ∈ V → ⦋𝐴 / 𝑥⦌(𝐵 ∪ 𝐶) = ∅)
21	16, 19, 20	3eqtr4rd 2867	. 2 ⊢ (¬ 𝐴 ∈ V → ⦋𝐴 / 𝑥⦌(𝐵 ∪ 𝐶) = (⦋𝐴 / 𝑥⦌𝐵 ∪ ⦋𝐴 / 𝑥⦌𝐶))
22	14, 21	pm2.61i 184	1 ⊢ ⦋𝐴 / 𝑥⦌(𝐵 ∪ 𝐶) = (⦋𝐴 / 𝑥⦌𝐵 ∪ ⦋𝐴 / 𝑥⦌𝐶)

Syntax hints:  ¬ wn 3   = wceq 1533   ∈ wcel 2110  Vcvv 3494  ⦋csb 3882   ∪ cun 3933  ∅c0 4290

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-fal 1546  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-nul 4291

This theorem is referenced by:  csbprg  4638