Theorem csbov123 7404

Description: Move class substitution in and out of an operation. (Contributed by NM, 12-Nov-2005.) (Revised by NM, 23-Aug-2018.)

Assertion

Ref	Expression
csbov123	⊢ ⦋𝐴 / 𝑥⦌(𝐵𝐹𝐶) = (⦋𝐴 / 𝑥⦌𝐵⦋𝐴 / 𝑥⦌𝐹⦋𝐴 / 𝑥⦌𝐶)

Proof of Theorem csbov123

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		csbeq1 3861	. . . 4 ⊢ (𝑦 = 𝐴 → ⦋𝑦 / 𝑥⦌(𝐵𝐹𝐶) = ⦋𝐴 / 𝑥⦌(𝐵𝐹𝐶))
2		csbeq1 3861	. . . . 5 ⊢ (𝑦 = 𝐴 → ⦋𝑦 / 𝑥⦌𝐹 = ⦋𝐴 / 𝑥⦌𝐹)
3		csbeq1 3861	. . . . 5 ⊢ (𝑦 = 𝐴 → ⦋𝑦 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐵)
4		csbeq1 3861	. . . . 5 ⊢ (𝑦 = 𝐴 → ⦋𝑦 / 𝑥⦌𝐶 = ⦋𝐴 / 𝑥⦌𝐶)
5	2, 3, 4	oveq123d 7383	. . . 4 ⊢ (𝑦 = 𝐴 → (⦋𝑦 / 𝑥⦌𝐵⦋𝑦 / 𝑥⦌𝐹⦋𝑦 / 𝑥⦌𝐶) = (⦋𝐴 / 𝑥⦌𝐵⦋𝐴 / 𝑥⦌𝐹⦋𝐴 / 𝑥⦌𝐶))
6	1, 5	eqeq12d 2747	. . 3 ⊢ (𝑦 = 𝐴 → (⦋𝑦 / 𝑥⦌(𝐵𝐹𝐶) = (⦋𝑦 / 𝑥⦌𝐵⦋𝑦 / 𝑥⦌𝐹⦋𝑦 / 𝑥⦌𝐶) ↔ ⦋𝐴 / 𝑥⦌(𝐵𝐹𝐶) = (⦋𝐴 / 𝑥⦌𝐵⦋𝐴 / 𝑥⦌𝐹⦋𝐴 / 𝑥⦌𝐶)))
7		vex 3450	. . . 4 ⊢ 𝑦 ∈ V
8		nfcsb1v 3883	. . . . 5 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐵
9		nfcsb1v 3883	. . . . 5 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐹
10		nfcsb1v 3883	. . . . 5 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐶
11	8, 9, 10	nfov 7392	. . . 4 ⊢ Ⅎ𝑥(⦋𝑦 / 𝑥⦌𝐵⦋𝑦 / 𝑥⦌𝐹⦋𝑦 / 𝑥⦌𝐶)
12		csbeq1a 3872	. . . . 5 ⊢ (𝑥 = 𝑦 → 𝐹 = ⦋𝑦 / 𝑥⦌𝐹)
13		csbeq1a 3872	. . . . 5 ⊢ (𝑥 = 𝑦 → 𝐵 = ⦋𝑦 / 𝑥⦌𝐵)
14		csbeq1a 3872	. . . . 5 ⊢ (𝑥 = 𝑦 → 𝐶 = ⦋𝑦 / 𝑥⦌𝐶)
15	12, 13, 14	oveq123d 7383	. . . 4 ⊢ (𝑥 = 𝑦 → (𝐵𝐹𝐶) = (⦋𝑦 / 𝑥⦌𝐵⦋𝑦 / 𝑥⦌𝐹⦋𝑦 / 𝑥⦌𝐶))
16	7, 11, 15	csbief 3893	. . 3 ⊢ ⦋𝑦 / 𝑥⦌(𝐵𝐹𝐶) = (⦋𝑦 / 𝑥⦌𝐵⦋𝑦 / 𝑥⦌𝐹⦋𝑦 / 𝑥⦌𝐶)
17	6, 16	vtoclg 3526	. 2 ⊢ (𝐴 ∈ V → ⦋𝐴 / 𝑥⦌(𝐵𝐹𝐶) = (⦋𝐴 / 𝑥⦌𝐵⦋𝐴 / 𝑥⦌𝐹⦋𝐴 / 𝑥⦌𝐶))
18		csbprc 4371	. . 3 ⊢ (¬ 𝐴 ∈ V → ⦋𝐴 / 𝑥⦌(𝐵𝐹𝐶) = ∅)
19		df-ov 7365	. . . 4 ⊢ (⦋𝐴 / 𝑥⦌𝐵⦋𝐴 / 𝑥⦌𝐹⦋𝐴 / 𝑥⦌𝐶) = (⦋𝐴 / 𝑥⦌𝐹‘⟨⦋𝐴 / 𝑥⦌𝐵, ⦋𝐴 / 𝑥⦌𝐶⟩)
20		csbprc 4371	. . . . . 6 ⊢ (¬ 𝐴 ∈ V → ⦋𝐴 / 𝑥⦌𝐹 = ∅)
21	20	fveq1d 6849	. . . . 5 ⊢ (¬ 𝐴 ∈ V → (⦋𝐴 / 𝑥⦌𝐹‘⟨⦋𝐴 / 𝑥⦌𝐵, ⦋𝐴 / 𝑥⦌𝐶⟩) = (∅‘⟨⦋𝐴 / 𝑥⦌𝐵, ⦋𝐴 / 𝑥⦌𝐶⟩))
22		0fv 6891	. . . . 5 ⊢ (∅‘⟨⦋𝐴 / 𝑥⦌𝐵, ⦋𝐴 / 𝑥⦌𝐶⟩) = ∅
23	21, 22	eqtrdi 2787	. . . 4 ⊢ (¬ 𝐴 ∈ V → (⦋𝐴 / 𝑥⦌𝐹‘⟨⦋𝐴 / 𝑥⦌𝐵, ⦋𝐴 / 𝑥⦌𝐶⟩) = ∅)
24	19, 23	eqtr2id 2784	. . 3 ⊢ (¬ 𝐴 ∈ V → ∅ = (⦋𝐴 / 𝑥⦌𝐵⦋𝐴 / 𝑥⦌𝐹⦋𝐴 / 𝑥⦌𝐶))
25	18, 24	eqtrd 2771	. 2 ⊢ (¬ 𝐴 ∈ V → ⦋𝐴 / 𝑥⦌(𝐵𝐹𝐶) = (⦋𝐴 / 𝑥⦌𝐵⦋𝐴 / 𝑥⦌𝐹⦋𝐴 / 𝑥⦌𝐶))
26	17, 25	pm2.61i 182	1 ⊢ ⦋𝐴 / 𝑥⦌(𝐵𝐹𝐶) = (⦋𝐴 / 𝑥⦌𝐵⦋𝐴 / 𝑥⦌𝐹⦋𝐴 / 𝑥⦌𝐶)

Syntax hints:  ¬ wn 3   = wceq 1541   ∈ wcel 2106  Vcvv 3446  ⦋csb 3858  ∅c0 4287  ⟨cop 4597  ‘cfv 6501  (class class class)co 7362

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-nul 5268  ax-pr 5389

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ral 3061  df-rex 3070  df-rab 3406  df-v 3448  df-sbc 3743  df-csb 3859  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4288  df-if 4492  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-br 5111  df-dm 5648  df-iota 6453  df-fv 6509  df-ov 7365

This theorem is referenced by:  csbov  7405  csbov12g  7406  csbfrecsg  8220  relowlpssretop  35908  rdgeqoa  35914