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Theorem csbov123 6802
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 3642 . . . 4 (𝑦 = 𝐴𝑦 / 𝑥(𝐵𝐹𝐶) = 𝐴 / 𝑥(𝐵𝐹𝐶))
2 csbeq1 3642 . . . . 5 (𝑦 = 𝐴𝑦 / 𝑥𝐹 = 𝐴 / 𝑥𝐹)
3 csbeq1 3642 . . . . 5 (𝑦 = 𝐴𝑦 / 𝑥𝐵 = 𝐴 / 𝑥𝐵)
4 csbeq1 3642 . . . . 5 (𝑦 = 𝐴𝑦 / 𝑥𝐶 = 𝐴 / 𝑥𝐶)
52, 3, 4oveq123d 6786 . . . 4 (𝑦 = 𝐴 → (𝑦 / 𝑥𝐵𝑦 / 𝑥𝐹𝑦 / 𝑥𝐶) = (𝐴 / 𝑥𝐵𝐴 / 𝑥𝐹𝐴 / 𝑥𝐶))
61, 5eqeq12d 2739 . . 3 (𝑦 = 𝐴 → (𝑦 / 𝑥(𝐵𝐹𝐶) = (𝑦 / 𝑥𝐵𝑦 / 𝑥𝐹𝑦 / 𝑥𝐶) ↔ 𝐴 / 𝑥(𝐵𝐹𝐶) = (𝐴 / 𝑥𝐵𝐴 / 𝑥𝐹𝐴 / 𝑥𝐶)))
7 vex 3307 . . . 4 𝑦 ∈ V
8 nfcsb1v 3655 . . . . 5 𝑥𝑦 / 𝑥𝐵
9 nfcsb1v 3655 . . . . 5 𝑥𝑦 / 𝑥𝐹
10 nfcsb1v 3655 . . . . 5 𝑥𝑦 / 𝑥𝐶
118, 9, 10nfov 6791 . . . 4 𝑥(𝑦 / 𝑥𝐵𝑦 / 𝑥𝐹𝑦 / 𝑥𝐶)
12 csbeq1a 3648 . . . . 5 (𝑥 = 𝑦𝐹 = 𝑦 / 𝑥𝐹)
13 csbeq1a 3648 . . . . 5 (𝑥 = 𝑦𝐵 = 𝑦 / 𝑥𝐵)
14 csbeq1a 3648 . . . . 5 (𝑥 = 𝑦𝐶 = 𝑦 / 𝑥𝐶)
1512, 13, 14oveq123d 6786 . . . 4 (𝑥 = 𝑦 → (𝐵𝐹𝐶) = (𝑦 / 𝑥𝐵𝑦 / 𝑥𝐹𝑦 / 𝑥𝐶))
167, 11, 15csbief 3664 . . 3 𝑦 / 𝑥(𝐵𝐹𝐶) = (𝑦 / 𝑥𝐵𝑦 / 𝑥𝐹𝑦 / 𝑥𝐶)
176, 16vtoclg 3370 . 2 (𝐴 ∈ V → 𝐴 / 𝑥(𝐵𝐹𝐶) = (𝐴 / 𝑥𝐵𝐴 / 𝑥𝐹𝐴 / 𝑥𝐶))
18 csbprc 4088 . . 3 𝐴 ∈ V → 𝐴 / 𝑥(𝐵𝐹𝐶) = ∅)
19 df-ov 6768 . . . 4 (𝐴 / 𝑥𝐵𝐴 / 𝑥𝐹𝐴 / 𝑥𝐶) = (𝐴 / 𝑥𝐹‘⟨𝐴 / 𝑥𝐵, 𝐴 / 𝑥𝐶⟩)
20 csbprc 4088 . . . . . 6 𝐴 ∈ V → 𝐴 / 𝑥𝐹 = ∅)
2120fveq1d 6306 . . . . 5 𝐴 ∈ V → (𝐴 / 𝑥𝐹‘⟨𝐴 / 𝑥𝐵, 𝐴 / 𝑥𝐶⟩) = (∅‘⟨𝐴 / 𝑥𝐵, 𝐴 / 𝑥𝐶⟩))
22 0fv 6340 . . . . 5 (∅‘⟨𝐴 / 𝑥𝐵, 𝐴 / 𝑥𝐶⟩) = ∅
2321, 22syl6eq 2774 . . . 4 𝐴 ∈ V → (𝐴 / 𝑥𝐹‘⟨𝐴 / 𝑥𝐵, 𝐴 / 𝑥𝐶⟩) = ∅)
2419, 23syl5req 2771 . . 3 𝐴 ∈ V → ∅ = (𝐴 / 𝑥𝐵𝐴 / 𝑥𝐹𝐴 / 𝑥𝐶))
2518, 24eqtrd 2758 . 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 1596  wcel 2103  Vcvv 3304  csb 3639  c0 4023  cop 4291  cfv 6001  (class class class)co 6765
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-8 2105  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704  ax-nul 4897  ax-pow 4948
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1599  df-fal 1602  df-ex 1818  df-nf 1823  df-sb 2011  df-eu 2575  df-mo 2576  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-ral 3019  df-rex 3020  df-rab 3023  df-v 3306  df-sbc 3542  df-csb 3640  df-dif 3683  df-un 3685  df-in 3687  df-ss 3694  df-nul 4024  df-if 4195  df-sn 4286  df-pr 4288  df-op 4292  df-uni 4545  df-br 4761  df-dm 5228  df-iota 5964  df-fv 6009  df-ov 6768
This theorem is referenced by:  csbov  6803  csbov12g  6804  relowlpssretop  33444  rdgeqoa  33450
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