Theorem csbuni 4900

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 4403	. . . 4 ⊢ ⦋𝐴 / 𝑥⦌{𝑧 ∣ ∃𝑦(𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵)} = {𝑧 ∣ [𝐴 / 𝑥]∃𝑦(𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵)}
2		sbcex2 3814	. . . . . 6 ⊢ ([𝐴 / 𝑥]∃𝑦(𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵) ↔ ∃𝑦[𝐴 / 𝑥](𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵))
3		sbcan 3803	. . . . . . . 8 ⊢ ([𝐴 / 𝑥](𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵) ↔ ([𝐴 / 𝑥]𝑧 ∈ 𝑦 ∧ [𝐴 / 𝑥]𝑦 ∈ 𝐵))
4		sbcg 3826	. . . . . . . . . 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 4381	. . . . . . . . . 10 ⊢ ([𝐴 / 𝑥]𝑦 ∈ 𝐵 ↔ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐵)
7	6	anbi2i 623	. . . . . . . . 9 ⊢ ((𝑧 ∈ 𝑦 ∧ [𝐴 / 𝑥]𝑦 ∈ 𝐵) ↔ (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐵))
8	5, 7	bitrdi 287	. . . . . . . 8 ⊢ (𝐴 ∈ V → (([𝐴 / 𝑥]𝑧 ∈ 𝑦 ∧ [𝐴 / 𝑥]𝑦 ∈ 𝐵) ↔ (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐵)))
9	3, 8	bitrid 283	. . . . . . 7 ⊢ (𝐴 ∈ V → ([𝐴 / 𝑥](𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵) ↔ (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐵)))
10	9	exbidv 1921	. . . . . 6 ⊢ (𝐴 ∈ V → (∃𝑦[𝐴 / 𝑥](𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵) ↔ ∃𝑦(𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐵)))
11	2, 10	bitrid 283	. . . . 5 ⊢ (𝐴 ∈ V → ([𝐴 / 𝑥]∃𝑦(𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵) ↔ ∃𝑦(𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐵)))
12	11	abbidv 2795	. . . 4 ⊢ (𝐴 ∈ V → {𝑧 ∣ [𝐴 / 𝑥]∃𝑦(𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵)} = {𝑧 ∣ ∃𝑦(𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐵)})
13	1, 12	eqtrid 2776	. . 3 ⊢ (𝐴 ∈ V → ⦋𝐴 / 𝑥⦌{𝑧 ∣ ∃𝑦(𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵)} = {𝑧 ∣ ∃𝑦(𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐵)})
14		df-uni 4872	. . . 4 ⊢ ∪ 𝐵 = {𝑧 ∣ ∃𝑦(𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵)}
15	14	csbeq2i 3870	. . 3 ⊢ ⦋𝐴 / 𝑥⦌∪ 𝐵 = ⦋𝐴 / 𝑥⦌{𝑧 ∣ ∃𝑦(𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵)}
16		df-uni 4872	. . 3 ⊢ ∪ ⦋𝐴 / 𝑥⦌𝐵 = {𝑧 ∣ ∃𝑦(𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐵)}
17	13, 15, 16	3eqtr4g 2789	. 2 ⊢ (𝐴 ∈ V → ⦋𝐴 / 𝑥⦌∪ 𝐵 = ∪ ⦋𝐴 / 𝑥⦌𝐵)
18		csbprc 4372	. . 3 ⊢ (¬ 𝐴 ∈ V → ⦋𝐴 / 𝑥⦌∪ 𝐵 = ∅)
19		csbprc 4372	. . . . 5 ⊢ (¬ 𝐴 ∈ V → ⦋𝐴 / 𝑥⦌𝐵 = ∅)
20	19	unieqd 4884	. . . 4 ⊢ (¬ 𝐴 ∈ V → ∪ ⦋𝐴 / 𝑥⦌𝐵 = ∪ ∅)
21		uni0 4899	. . . 4 ⊢ ∪ ∅ = ∅
22	20, 21	eqtr2di 2781	. . 3 ⊢ (¬ 𝐴 ∈ V → ∅ = ∪ ⦋𝐴 / 𝑥⦌𝐵)
23	18, 22	eqtrd 2764	. 2 ⊢ (¬ 𝐴 ∈ V → ⦋𝐴 / 𝑥⦌∪ 𝐵 = ∪ ⦋𝐴 / 𝑥⦌𝐵)
24	17, 23	pm2.61i 182	1 ⊢ ⦋𝐴 / 𝑥⦌∪ 𝐵 = ∪ ⦋𝐴 / 𝑥⦌𝐵

Syntax hints:  ¬ wn 3   ∧ wa 395   = wceq 1540  ∃wex 1779   ∈ wcel 2109  {cab 2707  Vcvv 3447  [wsbc 3753  ⦋csb 3862  ∅c0 4296  ∪ cuni 4871

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 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ral 3045  df-rex 3054  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-ss 3931  df-nul 4297  df-sn 4590  df-uni 4872

This theorem is referenced by:  csbfrecsg  8263  csbfv12gALTVD  44888