Theorem csbun 4434

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 3892	. . . 4 ⊢ (𝑦 = 𝐴 → ⦋𝑦 / 𝑥⦌(𝐵 ∪ 𝐶) = ⦋𝐴 / 𝑥⦌(𝐵 ∪ 𝐶))
2		csbeq1 3892	. . . . 5 ⊢ (𝑦 = 𝐴 → ⦋𝑦 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐵)
3		csbeq1 3892	. . . . 5 ⊢ (𝑦 = 𝐴 → ⦋𝑦 / 𝑥⦌𝐶 = ⦋𝐴 / 𝑥⦌𝐶)
4	2, 3	uneq12d 4160	. . . 4 ⊢ (𝑦 = 𝐴 → (⦋𝑦 / 𝑥⦌𝐵 ∪ ⦋𝑦 / 𝑥⦌𝐶) = (⦋𝐴 / 𝑥⦌𝐵 ∪ ⦋𝐴 / 𝑥⦌𝐶))
5	1, 4	eqeq12d 2743	. . 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 3473	. . . 4 ⊢ 𝑦 ∈ V
7		nfcsb1v 3914	. . . . 5 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐵
8		nfcsb1v 3914	. . . . 5 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐶
9	7, 8	nfun 4161	. . . 4 ⊢ Ⅎ𝑥(⦋𝑦 / 𝑥⦌𝐵 ∪ ⦋𝑦 / 𝑥⦌𝐶)
10		csbeq1a 3903	. . . . 5 ⊢ (𝑥 = 𝑦 → 𝐵 = ⦋𝑦 / 𝑥⦌𝐵)
11		csbeq1a 3903	. . . . 5 ⊢ (𝑥 = 𝑦 → 𝐶 = ⦋𝑦 / 𝑥⦌𝐶)
12	10, 11	uneq12d 4160	. . . 4 ⊢ (𝑥 = 𝑦 → (𝐵 ∪ 𝐶) = (⦋𝑦 / 𝑥⦌𝐵 ∪ ⦋𝑦 / 𝑥⦌𝐶))
13	6, 9, 12	csbief 3924	. . 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		un0 4386	. . . 4 ⊢ (∅ ∪ ∅) = ∅
16	15	a1i 11	. . 3 ⊢ (¬ 𝐴 ∈ V → (∅ ∪ ∅) = ∅)
17		csbprc 4402	. . . 4 ⊢ (¬ 𝐴 ∈ V → ⦋𝐴 / 𝑥⦌𝐵 = ∅)
18		csbprc 4402	. . . 4 ⊢ (¬ 𝐴 ∈ V → ⦋𝐴 / 𝑥⦌𝐶 = ∅)
19	17, 18	uneq12d 4160	. . 3 ⊢ (¬ 𝐴 ∈ V → (⦋𝐴 / 𝑥⦌𝐵 ∪ ⦋𝐴 / 𝑥⦌𝐶) = (∅ ∪ ∅))
20		csbprc 4402	. . 3 ⊢ (¬ 𝐴 ∈ V → ⦋𝐴 / 𝑥⦌(𝐵 ∪ 𝐶) = ∅)
21	16, 19, 20	3eqtr4rd 2778	. 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 1534   ∈ wcel 2099  Vcvv 3469  ⦋csb 3889   ∪ cun 3942  ∅c0 4318

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-nul 4319

This theorem is referenced by:  csbprg  4709