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Theorem csbov 6851
Description: Move class substitution in and out of an operation. (Contributed by NM, 23-Aug-2018.)
Assertion
Ref Expression
csbov 𝐴 / 𝑥(𝐵𝐹𝐶) = (𝐵𝐴 / 𝑥𝐹𝐶)
Distinct variable groups:   𝑥,𝐵   𝑥,𝐶
Allowed substitution hints:   𝐴(𝑥)   𝐹(𝑥)

Proof of Theorem csbov
StepHypRef Expression
1 csbov123 6850 . 2 𝐴 / 𝑥(𝐵𝐹𝐶) = (𝐴 / 𝑥𝐵𝐴 / 𝑥𝐹𝐴 / 𝑥𝐶)
2 csbconstg 3687 . . . 4 (𝐴 ∈ V → 𝐴 / 𝑥𝐵 = 𝐵)
3 csbconstg 3687 . . . 4 (𝐴 ∈ V → 𝐴 / 𝑥𝐶 = 𝐶)
42, 3oveq12d 6831 . . 3 (𝐴 ∈ V → (𝐴 / 𝑥𝐵𝐴 / 𝑥𝐹𝐴 / 𝑥𝐶) = (𝐵𝐴 / 𝑥𝐹𝐶))
5 0fv 6388 . . . . 5 (∅‘⟨𝐵, 𝐶⟩) = ∅
6 df-ov 6816 . . . . 5 (𝐵𝐶) = (∅‘⟨𝐵, 𝐶⟩)
7 df-ov 6816 . . . . . 6 (𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶) = (∅‘⟨𝐴 / 𝑥𝐵, 𝐴 / 𝑥𝐶⟩)
8 0fv 6388 . . . . . 6 (∅‘⟨𝐴 / 𝑥𝐵, 𝐴 / 𝑥𝐶⟩) = ∅
97, 8eqtri 2782 . . . . 5 (𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶) = ∅
105, 6, 93eqtr4ri 2793 . . . 4 (𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶) = (𝐵𝐶)
11 csbprc 4123 . . . . 5 𝐴 ∈ V → 𝐴 / 𝑥𝐹 = ∅)
1211oveqd 6830 . . . 4 𝐴 ∈ V → (𝐴 / 𝑥𝐵𝐴 / 𝑥𝐹𝐴 / 𝑥𝐶) = (𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶))
1311oveqd 6830 . . . 4 𝐴 ∈ V → (𝐵𝐴 / 𝑥𝐹𝐶) = (𝐵𝐶))
1410, 12, 133eqtr4a 2820 . . 3 𝐴 ∈ V → (𝐴 / 𝑥𝐵𝐴 / 𝑥𝐹𝐴 / 𝑥𝐶) = (𝐵𝐴 / 𝑥𝐹𝐶))
154, 14pm2.61i 176 . 2 (𝐴 / 𝑥𝐵𝐴 / 𝑥𝐹𝐴 / 𝑥𝐶) = (𝐵𝐴 / 𝑥𝐹𝐶)
161, 15eqtri 2782 1 𝐴 / 𝑥(𝐵𝐹𝐶) = (𝐵𝐴 / 𝑥𝐹𝐶)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1632  wcel 2139  Vcvv 3340  csb 3674  c0 4058  cop 4327  cfv 6049  (class class class)co 6813
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-nul 4941  ax-pow 4992
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-fal 1638  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-dm 5276  df-iota 6012  df-fv 6057  df-ov 6816
This theorem is referenced by:  mptcoe1matfsupp  20809  mp2pm2mplem4  20816
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