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Theorem csbov 7476
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 7475 . 2 𝐴 / 𝑥(𝐵𝐹𝐶) = (𝐴 / 𝑥𝐵𝐴 / 𝑥𝐹𝐴 / 𝑥𝐶)
2 csbconstg 3927 . . . 4 (𝐴 ∈ V → 𝐴 / 𝑥𝐵 = 𝐵)
3 csbconstg 3927 . . . 4 (𝐴 ∈ V → 𝐴 / 𝑥𝐶 = 𝐶)
42, 3oveq12d 7449 . . 3 (𝐴 ∈ V → (𝐴 / 𝑥𝐵𝐴 / 𝑥𝐹𝐴 / 𝑥𝐶) = (𝐵𝐴 / 𝑥𝐹𝐶))
5 0fv 6951 . . . . 5 (∅‘⟨𝐵, 𝐶⟩) = ∅
6 df-ov 7434 . . . . 5 (𝐵𝐶) = (∅‘⟨𝐵, 𝐶⟩)
7 0ov 7468 . . . . 5 (𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶) = ∅
85, 6, 73eqtr4ri 2774 . . . 4 (𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶) = (𝐵𝐶)
9 csbprc 4415 . . . . 5 𝐴 ∈ V → 𝐴 / 𝑥𝐹 = ∅)
109oveqd 7448 . . . 4 𝐴 ∈ V → (𝐴 / 𝑥𝐵𝐴 / 𝑥𝐹𝐴 / 𝑥𝐶) = (𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶))
119oveqd 7448 . . . 4 𝐴 ∈ V → (𝐵𝐴 / 𝑥𝐹𝐶) = (𝐵𝐶))
128, 10, 113eqtr4a 2801 . . 3 𝐴 ∈ V → (𝐴 / 𝑥𝐵𝐴 / 𝑥𝐹𝐴 / 𝑥𝐶) = (𝐵𝐴 / 𝑥𝐹𝐶))
134, 12pm2.61i 182 . 2 (𝐴 / 𝑥𝐵𝐴 / 𝑥𝐹𝐴 / 𝑥𝐶) = (𝐵𝐴 / 𝑥𝐹𝐶)
141, 13eqtri 2763 1 𝐴 / 𝑥(𝐵𝐹𝐶) = (𝐵𝐴 / 𝑥𝐹𝐶)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1537  wcel 2106  Vcvv 3478  csb 3908  c0 4339  cop 4637  cfv 6563  (class class class)co 7431
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-dm 5699  df-iota 6516  df-fv 6571  df-ov 7434
This theorem is referenced by:  mptcoe1matfsupp  22824  mp2pm2mplem4  22831
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