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Theorem csbov 7194
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 7193 . 2 𝐴 / 𝑥(𝐵𝐹𝐶) = (𝐴 / 𝑥𝐵𝐴 / 𝑥𝐹𝐴 / 𝑥𝐶)
2 csbconstg 3905 . . . 4 (𝐴 ∈ V → 𝐴 / 𝑥𝐵 = 𝐵)
3 csbconstg 3905 . . . 4 (𝐴 ∈ V → 𝐴 / 𝑥𝐶 = 𝐶)
42, 3oveq12d 7169 . . 3 (𝐴 ∈ V → (𝐴 / 𝑥𝐵𝐴 / 𝑥𝐹𝐴 / 𝑥𝐶) = (𝐵𝐴 / 𝑥𝐹𝐶))
5 0fv 6705 . . . . 5 (∅‘⟨𝐵, 𝐶⟩) = ∅
6 df-ov 7154 . . . . 5 (𝐵𝐶) = (∅‘⟨𝐵, 𝐶⟩)
7 0ov 7188 . . . . 5 (𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶) = ∅
85, 6, 73eqtr4ri 2859 . . . 4 (𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶) = (𝐵𝐶)
9 csbprc 4361 . . . . 5 𝐴 ∈ V → 𝐴 / 𝑥𝐹 = ∅)
109oveqd 7168 . . . 4 𝐴 ∈ V → (𝐴 / 𝑥𝐵𝐴 / 𝑥𝐹𝐴 / 𝑥𝐶) = (𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶))
119oveqd 7168 . . . 4 𝐴 ∈ V → (𝐵𝐴 / 𝑥𝐹𝐶) = (𝐵𝐶))
128, 10, 113eqtr4a 2886 . . 3 𝐴 ∈ V → (𝐴 / 𝑥𝐵𝐴 / 𝑥𝐹𝐴 / 𝑥𝐶) = (𝐵𝐴 / 𝑥𝐹𝐶))
134, 12pm2.61i 183 . 2 (𝐴 / 𝑥𝐵𝐴 / 𝑥𝐹𝐴 / 𝑥𝐶) = (𝐵𝐴 / 𝑥𝐹𝐶)
141, 13eqtri 2848 1 𝐴 / 𝑥(𝐵𝐹𝐶) = (𝐵𝐴 / 𝑥𝐹𝐶)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1530  wcel 2107  Vcvv 3499  csb 3886  c0 4294  cop 4569  cfv 6351  (class class class)co 7151
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2797  ax-nul 5206  ax-pow 5262
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-fal 1543  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2619  df-eu 2651  df-clab 2804  df-cleq 2818  df-clel 2897  df-nfc 2967  df-ral 3147  df-rex 3148  df-rab 3151  df-v 3501  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4470  df-sn 4564  df-pr 4566  df-op 4570  df-uni 4837  df-br 5063  df-dm 5563  df-iota 6311  df-fv 6359  df-ov 7154
This theorem is referenced by:  mptcoe1matfsupp  21329  mp2pm2mplem4  21336
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