Theorem csbuni 4859

Description: Distribute proper substitution through the union of a class. (Contributed by Alan Sare, 10-Nov-2012.) (Revised by NM, 22-Aug-2018.)

Assertion

Ref	Expression
csbuni	⊢ ⦋𝐴 / 𝑥⦌∪ 𝐵 = ∪ ⦋𝐴 / 𝑥⦌𝐵

Proof of Theorem csbuni

Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		csbab 4388	. . . 4 ⊢ ⦋𝐴 / 𝑥⦌{𝑧 ∣ ∃𝑦(𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵)} = {𝑧 ∣ [𝐴 / 𝑥]∃𝑦(𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵)}
2		sbcex2 3833	. . . . . 6 ⊢ ([𝐴 / 𝑥]∃𝑦(𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵) ↔ ∃𝑦[𝐴 / 𝑥](𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵))
3		sbcan 3820	. . . . . . . 8 ⊢ ([𝐴 / 𝑥](𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵) ↔ ([𝐴 / 𝑥]𝑧 ∈ 𝑦 ∧ [𝐴 / 𝑥]𝑦 ∈ 𝐵))
4		sbcg 3846	. . . . . . . . . 10 ⊢ (𝐴 ∈ V → ([𝐴 / 𝑥]𝑧 ∈ 𝑦 ↔ 𝑧 ∈ 𝑦))
5	4	anbi1d 631	. . . . . . . . 9 ⊢ (𝐴 ∈ V → (([𝐴 / 𝑥]𝑧 ∈ 𝑦 ∧ [𝐴 / 𝑥]𝑦 ∈ 𝐵) ↔ (𝑧 ∈ 𝑦 ∧ [𝐴 / 𝑥]𝑦 ∈ 𝐵)))
6		sbcel2 4366	. . . . . . . . . 10 ⊢ ([𝐴 / 𝑥]𝑦 ∈ 𝐵 ↔ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐵)
7	6	anbi2i 624	. . . . . . . . 9 ⊢ ((𝑧 ∈ 𝑦 ∧ [𝐴 / 𝑥]𝑦 ∈ 𝐵) ↔ (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐵))
8	5, 7	syl6bb 289	. . . . . . . 8 ⊢ (𝐴 ∈ V → (([𝐴 / 𝑥]𝑧 ∈ 𝑦 ∧ [𝐴 / 𝑥]𝑦 ∈ 𝐵) ↔ (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐵)))
9	3, 8	syl5bb 285	. . . . . . 7 ⊢ (𝐴 ∈ V → ([𝐴 / 𝑥](𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵) ↔ (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐵)))
10	9	exbidv 1918	. . . . . 6 ⊢ (𝐴 ∈ V → (∃𝑦[𝐴 / 𝑥](𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵) ↔ ∃𝑦(𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐵)))
11	2, 10	syl5bb 285	. . . . 5 ⊢ (𝐴 ∈ V → ([𝐴 / 𝑥]∃𝑦(𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵) ↔ ∃𝑦(𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐵)))
12	11	abbidv 2885	. . . 4 ⊢ (𝐴 ∈ V → {𝑧 ∣ [𝐴 / 𝑥]∃𝑦(𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵)} = {𝑧 ∣ ∃𝑦(𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐵)})
13	1, 12	syl5eq 2868	. . 3 ⊢ (𝐴 ∈ V → ⦋𝐴 / 𝑥⦌{𝑧 ∣ ∃𝑦(𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵)} = {𝑧 ∣ ∃𝑦(𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐵)})
14		df-uni 4832	. . . 4 ⊢ ∪ 𝐵 = {𝑧 ∣ ∃𝑦(𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵)}
15	14	csbeq2i 3890	. . 3 ⊢ ⦋𝐴 / 𝑥⦌∪ 𝐵 = ⦋𝐴 / 𝑥⦌{𝑧 ∣ ∃𝑦(𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵)}
16		df-uni 4832	. . 3 ⊢ ∪ ⦋𝐴 / 𝑥⦌𝐵 = {𝑧 ∣ ∃𝑦(𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐵)}
17	13, 15, 16	3eqtr4g 2881	. 2 ⊢ (𝐴 ∈ V → ⦋𝐴 / 𝑥⦌∪ 𝐵 = ∪ ⦋𝐴 / 𝑥⦌𝐵)
18		csbprc 4357	. . 3 ⊢ (¬ 𝐴 ∈ V → ⦋𝐴 / 𝑥⦌∪ 𝐵 = ∅)
19		csbprc 4357	. . . . 5 ⊢ (¬ 𝐴 ∈ V → ⦋𝐴 / 𝑥⦌𝐵 = ∅)
20	19	unieqd 4841	. . . 4 ⊢ (¬ 𝐴 ∈ V → ∪ ⦋𝐴 / 𝑥⦌𝐵 = ∪ ∅)
21		uni0 4858	. . . 4 ⊢ ∪ ∅ = ∅
22	20, 21	syl6req 2873	. . 3 ⊢ (¬ 𝐴 ∈ V → ∅ = ∪ ⦋𝐴 / 𝑥⦌𝐵)
23	18, 22	eqtrd 2856	. 2 ⊢ (¬ 𝐴 ∈ V → ⦋𝐴 / 𝑥⦌∪ 𝐵 = ∪ ⦋𝐴 / 𝑥⦌𝐵)
24	17, 23	pm2.61i 184	1 ⊢ ⦋𝐴 / 𝑥⦌∪ 𝐵 = ∪ ⦋𝐴 / 𝑥⦌𝐵

Syntax hints:  ¬ wn 3   ∧ wa 398   = wceq 1533  ∃wex 1776   ∈ wcel 2110  {cab 2799  Vcvv 3494  [wsbc 3771  ⦋csb 3882  ∅c0 4290  ∪ cuni 4831

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-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-ral 3143  df-rex 3144  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-in 3942  df-ss 3951  df-nul 4291  df-sn 4561  df-uni 4832

This theorem is referenced by:  csbwrecsg  34602  csbfv12gALTVD  41226