Theorem csbov123 7198

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 3810	. . . 4 ⊢ (𝑦 = 𝐴 → ⦋𝑦 / 𝑥⦌(𝐵𝐹𝐶) = ⦋𝐴 / 𝑥⦌(𝐵𝐹𝐶))
2		csbeq1 3810	. . . . 5 ⊢ (𝑦 = 𝐴 → ⦋𝑦 / 𝑥⦌𝐹 = ⦋𝐴 / 𝑥⦌𝐹)
3		csbeq1 3810	. . . . 5 ⊢ (𝑦 = 𝐴 → ⦋𝑦 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐵)
4		csbeq1 3810	. . . . 5 ⊢ (𝑦 = 𝐴 → ⦋𝑦 / 𝑥⦌𝐶 = ⦋𝐴 / 𝑥⦌𝐶)
5	2, 3, 4	oveq123d 7177	. . . 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 2774	. . 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 3413	. . . 4 ⊢ 𝑦 ∈ V
8		nfcsb1v 3831	. . . . 5 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐵
9		nfcsb1v 3831	. . . . 5 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐹
10		nfcsb1v 3831	. . . . 5 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐶
11	8, 9, 10	nfov 7186	. . . 4 ⊢ Ⅎ𝑥(⦋𝑦 / 𝑥⦌𝐵⦋𝑦 / 𝑥⦌𝐹⦋𝑦 / 𝑥⦌𝐶)
12		csbeq1a 3821	. . . . 5 ⊢ (𝑥 = 𝑦 → 𝐹 = ⦋𝑦 / 𝑥⦌𝐹)
13		csbeq1a 3821	. . . . 5 ⊢ (𝑥 = 𝑦 → 𝐵 = ⦋𝑦 / 𝑥⦌𝐵)
14		csbeq1a 3821	. . . . 5 ⊢ (𝑥 = 𝑦 → 𝐶 = ⦋𝑦 / 𝑥⦌𝐶)
15	12, 13, 14	oveq123d 7177	. . . 4 ⊢ (𝑥 = 𝑦 → (𝐵𝐹𝐶) = (⦋𝑦 / 𝑥⦌𝐵⦋𝑦 / 𝑥⦌𝐹⦋𝑦 / 𝑥⦌𝐶))
16	7, 11, 15	csbief 3841	. . 3 ⊢ ⦋𝑦 / 𝑥⦌(𝐵𝐹𝐶) = (⦋𝑦 / 𝑥⦌𝐵⦋𝑦 / 𝑥⦌𝐹⦋𝑦 / 𝑥⦌𝐶)
17	6, 16	vtoclg 3487	. 2 ⊢ (𝐴 ∈ V → ⦋𝐴 / 𝑥⦌(𝐵𝐹𝐶) = (⦋𝐴 / 𝑥⦌𝐵⦋𝐴 / 𝑥⦌𝐹⦋𝐴 / 𝑥⦌𝐶))
18		csbprc 4305	. . 3 ⊢ (¬ 𝐴 ∈ V → ⦋𝐴 / 𝑥⦌(𝐵𝐹𝐶) = ∅)
19		df-ov 7159	. . . 4 ⊢ (⦋𝐴 / 𝑥⦌𝐵⦋𝐴 / 𝑥⦌𝐹⦋𝐴 / 𝑥⦌𝐶) = (⦋𝐴 / 𝑥⦌𝐹‘⟨⦋𝐴 / 𝑥⦌𝐵, ⦋𝐴 / 𝑥⦌𝐶⟩)
20		csbprc 4305	. . . . . 6 ⊢ (¬ 𝐴 ∈ V → ⦋𝐴 / 𝑥⦌𝐹 = ∅)
21	20	fveq1d 6665	. . . . 5 ⊢ (¬ 𝐴 ∈ V → (⦋𝐴 / 𝑥⦌𝐹‘⟨⦋𝐴 / 𝑥⦌𝐵, ⦋𝐴 / 𝑥⦌𝐶⟩) = (∅‘⟨⦋𝐴 / 𝑥⦌𝐵, ⦋𝐴 / 𝑥⦌𝐶⟩))
22		0fv 6702	. . . . 5 ⊢ (∅‘⟨⦋𝐴 / 𝑥⦌𝐵, ⦋𝐴 / 𝑥⦌𝐶⟩) = ∅
23	21, 22	eqtrdi 2809	. . . 4 ⊢ (¬ 𝐴 ∈ V → (⦋𝐴 / 𝑥⦌𝐹‘⟨⦋𝐴 / 𝑥⦌𝐵, ⦋𝐴 / 𝑥⦌𝐶⟩) = ∅)
24	19, 23	syl5req 2806	. . 3 ⊢ (¬ 𝐴 ∈ V → ∅ = (⦋𝐴 / 𝑥⦌𝐵⦋𝐴 / 𝑥⦌𝐹⦋𝐴 / 𝑥⦌𝐶))
25	18, 24	eqtrd 2793	. 2 ⊢ (¬ 𝐴 ∈ V → ⦋𝐴 / 𝑥⦌(𝐵𝐹𝐶) = (⦋𝐴 / 𝑥⦌𝐵⦋𝐴 / 𝑥⦌𝐹⦋𝐴 / 𝑥⦌𝐶))
26	17, 25	pm2.61i 185	1 ⊢ ⦋𝐴 / 𝑥⦌(𝐵𝐹𝐶) = (⦋𝐴 / 𝑥⦌𝐵⦋𝐴 / 𝑥⦌𝐹⦋𝐴 / 𝑥⦌𝐶)

Syntax hints:  ¬ wn 3   = wceq 1538   ∈ wcel 2111  Vcvv 3409  ⦋csb 3807  ∅c0 4227  ⟨cop 4531  ‘cfv 6340  (class class class)co 7156

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5173  ax-nul 5180  ax-pr 5302

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-v 3411  df-sbc 3699  df-csb 3808  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-nul 4228  df-if 4424  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-br 5037  df-dm 5538  df-iota 6299  df-fv 6348  df-ov 7159

This theorem is referenced by:  csbov  7199  csbov12g  7200  relowlpssretop  35096  rdgeqoa  35102