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Theorem csbov123 6640
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 3517 . . . 4 (𝑦 = 𝐴𝑦 / 𝑥(𝐵𝐹𝐶) = 𝐴 / 𝑥(𝐵𝐹𝐶))
2 csbeq1 3517 . . . . 5 (𝑦 = 𝐴𝑦 / 𝑥𝐹 = 𝐴 / 𝑥𝐹)
3 csbeq1 3517 . . . . 5 (𝑦 = 𝐴𝑦 / 𝑥𝐵 = 𝐴 / 𝑥𝐵)
4 csbeq1 3517 . . . . 5 (𝑦 = 𝐴𝑦 / 𝑥𝐶 = 𝐴 / 𝑥𝐶)
52, 3, 4oveq123d 6625 . . . 4 (𝑦 = 𝐴 → (𝑦 / 𝑥𝐵𝑦 / 𝑥𝐹𝑦 / 𝑥𝐶) = (𝐴 / 𝑥𝐵𝐴 / 𝑥𝐹𝐴 / 𝑥𝐶))
61, 5eqeq12d 2636 . . 3 (𝑦 = 𝐴 → (𝑦 / 𝑥(𝐵𝐹𝐶) = (𝑦 / 𝑥𝐵𝑦 / 𝑥𝐹𝑦 / 𝑥𝐶) ↔ 𝐴 / 𝑥(𝐵𝐹𝐶) = (𝐴 / 𝑥𝐵𝐴 / 𝑥𝐹𝐴 / 𝑥𝐶)))
7 vex 3189 . . . 4 𝑦 ∈ V
8 nfcsb1v 3530 . . . . 5 𝑥𝑦 / 𝑥𝐵
9 nfcsb1v 3530 . . . . 5 𝑥𝑦 / 𝑥𝐹
10 nfcsb1v 3530 . . . . 5 𝑥𝑦 / 𝑥𝐶
118, 9, 10nfov 6630 . . . 4 𝑥(𝑦 / 𝑥𝐵𝑦 / 𝑥𝐹𝑦 / 𝑥𝐶)
12 csbeq1a 3523 . . . . 5 (𝑥 = 𝑦𝐹 = 𝑦 / 𝑥𝐹)
13 csbeq1a 3523 . . . . 5 (𝑥 = 𝑦𝐵 = 𝑦 / 𝑥𝐵)
14 csbeq1a 3523 . . . . 5 (𝑥 = 𝑦𝐶 = 𝑦 / 𝑥𝐶)
1512, 13, 14oveq123d 6625 . . . 4 (𝑥 = 𝑦 → (𝐵𝐹𝐶) = (𝑦 / 𝑥𝐵𝑦 / 𝑥𝐹𝑦 / 𝑥𝐶))
167, 11, 15csbief 3539 . . 3 𝑦 / 𝑥(𝐵𝐹𝐶) = (𝑦 / 𝑥𝐵𝑦 / 𝑥𝐹𝑦 / 𝑥𝐶)
176, 16vtoclg 3252 . 2 (𝐴 ∈ V → 𝐴 / 𝑥(𝐵𝐹𝐶) = (𝐴 / 𝑥𝐵𝐴 / 𝑥𝐹𝐴 / 𝑥𝐶))
18 csbprc 3952 . . 3 𝐴 ∈ V → 𝐴 / 𝑥(𝐵𝐹𝐶) = ∅)
19 df-ov 6607 . . . 4 (𝐴 / 𝑥𝐵𝐴 / 𝑥𝐹𝐴 / 𝑥𝐶) = (𝐴 / 𝑥𝐹‘⟨𝐴 / 𝑥𝐵, 𝐴 / 𝑥𝐶⟩)
20 csbprc 3952 . . . . . 6 𝐴 ∈ V → 𝐴 / 𝑥𝐹 = ∅)
2120fveq1d 6150 . . . . 5 𝐴 ∈ V → (𝐴 / 𝑥𝐹‘⟨𝐴 / 𝑥𝐵, 𝐴 / 𝑥𝐶⟩) = (∅‘⟨𝐴 / 𝑥𝐵, 𝐴 / 𝑥𝐶⟩))
22 0fv 6184 . . . . 5 (∅‘⟨𝐴 / 𝑥𝐵, 𝐴 / 𝑥𝐶⟩) = ∅
2321, 22syl6eq 2671 . . . 4 𝐴 ∈ V → (𝐴 / 𝑥𝐹‘⟨𝐴 / 𝑥𝐵, 𝐴 / 𝑥𝐶⟩) = ∅)
2419, 23syl5req 2668 . . 3 𝐴 ∈ V → ∅ = (𝐴 / 𝑥𝐵𝐴 / 𝑥𝐹𝐴 / 𝑥𝐶))
2518, 24eqtrd 2655 . 2 𝐴 ∈ V → 𝐴 / 𝑥(𝐵𝐹𝐶) = (𝐴 / 𝑥𝐵𝐴 / 𝑥𝐹𝐴 / 𝑥𝐶))
2617, 25pm2.61i 176 1 𝐴 / 𝑥(𝐵𝐹𝐶) = (𝐴 / 𝑥𝐵𝐴 / 𝑥𝐹𝐴 / 𝑥𝐶)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1480  wcel 1987  Vcvv 3186  csb 3514  c0 3891  cop 4154  cfv 5847  (class class class)co 6604
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-nul 4749  ax-pow 4803
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-fal 1486  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-dm 5084  df-iota 5810  df-fv 5855  df-ov 6607
This theorem is referenced by:  csbov  6641  csbov12g  6642  relowlpssretop  32841  rdgeqoa  32847
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