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 3851	. . . 4 ⊢ (𝑦 = 𝐴 → ⦋𝑦 / 𝑥⦌(𝐵 ∪ 𝐶) = ⦋𝐴 / 𝑥⦌(𝐵 ∪ 𝐶))
2		csbeq1 3851	. . . . 5 ⊢ (𝑦 = 𝐴 → ⦋𝑦 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐵)
3		csbeq1 3851	. . . . 5 ⊢ (𝑦 = 𝐴 → ⦋𝑦 / 𝑥⦌𝐶 = ⦋𝐴 / 𝑥⦌𝐶)
4	2, 3	uneq12d 4117	. . . 4 ⊢ (𝑦 = 𝐴 → (⦋𝑦 / 𝑥⦌𝐵 ∪ ⦋𝑦 / 𝑥⦌𝐶) = (⦋𝐴 / 𝑥⦌𝐵 ∪ ⦋𝐴 / 𝑥⦌𝐶))
5	1, 4	eqeq12d 2746	. . 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 3438	. . . 4 ⊢ 𝑦 ∈ V
7		nfcsb1v 3872	. . . . 5 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐵
8		nfcsb1v 3872	. . . . 5 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐶
9	7, 8	nfun 4118	. . . 4 ⊢ Ⅎ𝑥(⦋𝑦 / 𝑥⦌𝐵 ∪ ⦋𝑦 / 𝑥⦌𝐶)
10		csbeq1a 3862	. . . . 5 ⊢ (𝑥 = 𝑦 → 𝐵 = ⦋𝑦 / 𝑥⦌𝐵)
11		csbeq1a 3862	. . . . 5 ⊢ (𝑥 = 𝑦 → 𝐶 = ⦋𝑦 / 𝑥⦌𝐶)
12	10, 11	uneq12d 4117	. . . 4 ⊢ (𝑥 = 𝑦 → (𝐵 ∪ 𝐶) = (⦋𝑦 / 𝑥⦌𝐵 ∪ ⦋𝑦 / 𝑥⦌𝐶))
13	6, 9, 12	csbief 3882	. . 3 ⊢ ⦋𝑦 / 𝑥⦌(𝐵 ∪ 𝐶) = (⦋𝑦 / 𝑥⦌𝐵 ∪ ⦋𝑦 / 𝑥⦌𝐶)
14	5, 13	vtoclg 3507	. 2 ⊢ (𝐴 ∈ V → ⦋𝐴 / 𝑥⦌(𝐵 ∪ 𝐶) = (⦋𝐴 / 𝑥⦌𝐵 ∪ ⦋𝐴 / 𝑥⦌𝐶))
15		un0 4342	. . . 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 4117	. . 3 ⊢ (¬ 𝐴 ∈ V → (⦋𝐴 / 𝑥⦌𝐵 ∪ ⦋𝐴 / 𝑥⦌𝐶) = (∅ ∪ ∅))
20		csbprc 4357	. . 3 ⊢ (¬ 𝐴 ∈ V → ⦋𝐴 / 𝑥⦌(𝐵 ∪ 𝐶) = ∅)
21	16, 19, 20	3eqtr4rd 2776	. 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 1541   ∈ wcel 2110  Vcvv 3434  ⦋csb 3848   ∪ cun 3898  ∅c0 4281

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2112  ax-9 2120  ax-10 2143  ax-11 2159  ax-12 2179  ax-ext 2702

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2067  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-v 3436  df-sbc 3740  df-csb 3849  df-dif 3903  df-un 3905  df-nul 4282

This theorem is referenced by:  csbprg  4660