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Theorem csbov123 7400
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 / 𝑥𝐵𝐴 / 𝑥𝐹𝐴 / 𝑥𝐶)

Proof of Theorem csbov123
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 csbeq1 3859 . . . 4 (𝑦 = 𝐴𝑦 / 𝑥(𝐵𝐹𝐶) = 𝐴 / 𝑥(𝐵𝐹𝐶))
2 csbeq1 3859 . . . . 5 (𝑦 = 𝐴𝑦 / 𝑥𝐹 = 𝐴 / 𝑥𝐹)
3 csbeq1 3859 . . . . 5 (𝑦 = 𝐴𝑦 / 𝑥𝐵 = 𝐴 / 𝑥𝐵)
4 csbeq1 3859 . . . . 5 (𝑦 = 𝐴𝑦 / 𝑥𝐶 = 𝐴 / 𝑥𝐶)
52, 3, 4oveq123d 7379 . . . 4 (𝑦 = 𝐴 → (𝑦 / 𝑥𝐵𝑦 / 𝑥𝐹𝑦 / 𝑥𝐶) = (𝐴 / 𝑥𝐵𝐴 / 𝑥𝐹𝐴 / 𝑥𝐶))
61, 5eqeq12d 2753 . . 3 (𝑦 = 𝐴 → (𝑦 / 𝑥(𝐵𝐹𝐶) = (𝑦 / 𝑥𝐵𝑦 / 𝑥𝐹𝑦 / 𝑥𝐶) ↔ 𝐴 / 𝑥(𝐵𝐹𝐶) = (𝐴 / 𝑥𝐵𝐴 / 𝑥𝐹𝐴 / 𝑥𝐶)))
7 vex 3450 . . . 4 𝑦 ∈ V
8 nfcsb1v 3881 . . . . 5 𝑥𝑦 / 𝑥𝐵
9 nfcsb1v 3881 . . . . 5 𝑥𝑦 / 𝑥𝐹
10 nfcsb1v 3881 . . . . 5 𝑥𝑦 / 𝑥𝐶
118, 9, 10nfov 7388 . . . 4 𝑥(𝑦 / 𝑥𝐵𝑦 / 𝑥𝐹𝑦 / 𝑥𝐶)
12 csbeq1a 3870 . . . . 5 (𝑥 = 𝑦𝐹 = 𝑦 / 𝑥𝐹)
13 csbeq1a 3870 . . . . 5 (𝑥 = 𝑦𝐵 = 𝑦 / 𝑥𝐵)
14 csbeq1a 3870 . . . . 5 (𝑥 = 𝑦𝐶 = 𝑦 / 𝑥𝐶)
1512, 13, 14oveq123d 7379 . . . 4 (𝑥 = 𝑦 → (𝐵𝐹𝐶) = (𝑦 / 𝑥𝐵𝑦 / 𝑥𝐹𝑦 / 𝑥𝐶))
167, 11, 15csbief 3891 . . 3 𝑦 / 𝑥(𝐵𝐹𝐶) = (𝑦 / 𝑥𝐵𝑦 / 𝑥𝐹𝑦 / 𝑥𝐶)
176, 16vtoclg 3526 . 2 (𝐴 ∈ V → 𝐴 / 𝑥(𝐵𝐹𝐶) = (𝐴 / 𝑥𝐵𝐴 / 𝑥𝐹𝐴 / 𝑥𝐶))
18 csbprc 4367 . . 3 𝐴 ∈ V → 𝐴 / 𝑥(𝐵𝐹𝐶) = ∅)
19 df-ov 7361 . . . 4 (𝐴 / 𝑥𝐵𝐴 / 𝑥𝐹𝐴 / 𝑥𝐶) = (𝐴 / 𝑥𝐹‘⟨𝐴 / 𝑥𝐵, 𝐴 / 𝑥𝐶⟩)
20 csbprc 4367 . . . . . 6 𝐴 ∈ V → 𝐴 / 𝑥𝐹 = ∅)
2120fveq1d 6845 . . . . 5 𝐴 ∈ V → (𝐴 / 𝑥𝐹‘⟨𝐴 / 𝑥𝐵, 𝐴 / 𝑥𝐶⟩) = (∅‘⟨𝐴 / 𝑥𝐵, 𝐴 / 𝑥𝐶⟩))
22 0fv 6887 . . . . 5 (∅‘⟨𝐴 / 𝑥𝐵, 𝐴 / 𝑥𝐶⟩) = ∅
2321, 22eqtrdi 2793 . . . 4 𝐴 ∈ V → (𝐴 / 𝑥𝐹‘⟨𝐴 / 𝑥𝐵, 𝐴 / 𝑥𝐶⟩) = ∅)
2419, 23eqtr2id 2790 . . 3 𝐴 ∈ V → ∅ = (𝐴 / 𝑥𝐵𝐴 / 𝑥𝐹𝐴 / 𝑥𝐶))
2518, 24eqtrd 2777 . 2 𝐴 ∈ V → 𝐴 / 𝑥(𝐵𝐹𝐶) = (𝐴 / 𝑥𝐵𝐴 / 𝑥𝐹𝐴 / 𝑥𝐶))
2617, 25pm2.61i 182 1 𝐴 / 𝑥(𝐵𝐹𝐶) = (𝐴 / 𝑥𝐵𝐴 / 𝑥𝐹𝐴 / 𝑥𝐶)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1542  wcel 2107  Vcvv 3446  csb 3856  c0 4283  cop 4593  cfv 6497  (class class class)co 7358
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-nul 5264  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ral 3066  df-rex 3075  df-rab 3409  df-v 3448  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-dm 5644  df-iota 6449  df-fv 6505  df-ov 7361
This theorem is referenced by:  csbov  7401  csbov12g  7402  csbfrecsg  8216  relowlpssretop  35838  rdgeqoa  35844
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