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Theorem csbov 6837
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 6836 . 2 𝐴 / 𝑥(𝐵𝐹𝐶) = (𝐴 / 𝑥𝐵𝐴 / 𝑥𝐹𝐴 / 𝑥𝐶)
2 csbconstg 3695 . . . 4 (𝐴 ∈ V → 𝐴 / 𝑥𝐵 = 𝐵)
3 csbconstg 3695 . . . 4 (𝐴 ∈ V → 𝐴 / 𝑥𝐶 = 𝐶)
42, 3oveq12d 6814 . . 3 (𝐴 ∈ V → (𝐴 / 𝑥𝐵𝐴 / 𝑥𝐹𝐴 / 𝑥𝐶) = (𝐵𝐴 / 𝑥𝐹𝐶))
5 0fv 6370 . . . . 5 (∅‘⟨𝐵, 𝐶⟩) = ∅
6 df-ov 6799 . . . . 5 (𝐵𝐶) = (∅‘⟨𝐵, 𝐶⟩)
7 df-ov 6799 . . . . . 6 (𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶) = (∅‘⟨𝐴 / 𝑥𝐵, 𝐴 / 𝑥𝐶⟩)
8 0fv 6370 . . . . . 6 (∅‘⟨𝐴 / 𝑥𝐵, 𝐴 / 𝑥𝐶⟩) = ∅
97, 8eqtri 2793 . . . . 5 (𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶) = ∅
105, 6, 93eqtr4ri 2804 . . . 4 (𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶) = (𝐵𝐶)
11 csbprc 4125 . . . . 5 𝐴 ∈ V → 𝐴 / 𝑥𝐹 = ∅)
1211oveqd 6813 . . . 4 𝐴 ∈ V → (𝐴 / 𝑥𝐵𝐴 / 𝑥𝐹𝐴 / 𝑥𝐶) = (𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶))
1311oveqd 6813 . . . 4 𝐴 ∈ V → (𝐵𝐴 / 𝑥𝐹𝐶) = (𝐵𝐶))
1410, 12, 133eqtr4a 2831 . . 3 𝐴 ∈ V → (𝐴 / 𝑥𝐵𝐴 / 𝑥𝐹𝐴 / 𝑥𝐶) = (𝐵𝐴 / 𝑥𝐹𝐶))
154, 14pm2.61i 176 . 2 (𝐴 / 𝑥𝐵𝐴 / 𝑥𝐹𝐴 / 𝑥𝐶) = (𝐵𝐴 / 𝑥𝐹𝐶)
161, 15eqtri 2793 1 𝐴 / 𝑥(𝐵𝐹𝐶) = (𝐵𝐴 / 𝑥𝐹𝐶)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1631  wcel 2145  Vcvv 3351  csb 3682  c0 4063  cop 4323  cfv 6030  (class class class)co 6796
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-nul 4924  ax-pow 4975
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-fal 1637  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-br 4788  df-dm 5260  df-iota 5993  df-fv 6038  df-ov 6799
This theorem is referenced by:  mptcoe1matfsupp  20827  mp2pm2mplem4  20834
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