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Theorem csbopab 4932
 Description: Move substitution into a class abstraction. Version of csbopabgALT 4933 without a sethood antecedent but depending on more axioms. (Contributed by NM, 6-Aug-2007.) (Revised by NM, 23-Aug-2018.)
Assertion
Ref Expression
csbopab 𝐴 / 𝑥{⟨𝑦, 𝑧⟩ ∣ 𝜑} = {⟨𝑦, 𝑧⟩ ∣ [𝐴 / 𝑥]𝜑}
Distinct variable groups:   𝑦,𝑧,𝐴   𝑥,𝑦,𝑧
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)   𝐴(𝑥)

Proof of Theorem csbopab
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 csbeq1 3502 . . . 4 (𝑤 = 𝐴𝑤 / 𝑥{⟨𝑦, 𝑧⟩ ∣ 𝜑} = 𝐴 / 𝑥{⟨𝑦, 𝑧⟩ ∣ 𝜑})
2 dfsbcq2 3405 . . . . 5 (𝑤 = 𝐴 → ([𝑤 / 𝑥]𝜑[𝐴 / 𝑥]𝜑))
32opabbidv 4648 . . . 4 (𝑤 = 𝐴 → {⟨𝑦, 𝑧⟩ ∣ [𝑤 / 𝑥]𝜑} = {⟨𝑦, 𝑧⟩ ∣ [𝐴 / 𝑥]𝜑})
41, 3eqeq12d 2625 . . 3 (𝑤 = 𝐴 → (𝑤 / 𝑥{⟨𝑦, 𝑧⟩ ∣ 𝜑} = {⟨𝑦, 𝑧⟩ ∣ [𝑤 / 𝑥]𝜑} ↔ 𝐴 / 𝑥{⟨𝑦, 𝑧⟩ ∣ 𝜑} = {⟨𝑦, 𝑧⟩ ∣ [𝐴 / 𝑥]𝜑}))
5 vex 3176 . . . 4 𝑤 ∈ V
6 nfs1v 2425 . . . . 5 𝑥[𝑤 / 𝑥]𝜑
76nfopab 4650 . . . 4 𝑥{⟨𝑦, 𝑧⟩ ∣ [𝑤 / 𝑥]𝜑}
8 sbequ12 2097 . . . . 5 (𝑥 = 𝑤 → (𝜑 ↔ [𝑤 / 𝑥]𝜑))
98opabbidv 4648 . . . 4 (𝑥 = 𝑤 → {⟨𝑦, 𝑧⟩ ∣ 𝜑} = {⟨𝑦, 𝑧⟩ ∣ [𝑤 / 𝑥]𝜑})
105, 7, 9csbief 3524 . . 3 𝑤 / 𝑥{⟨𝑦, 𝑧⟩ ∣ 𝜑} = {⟨𝑦, 𝑧⟩ ∣ [𝑤 / 𝑥]𝜑}
114, 10vtoclg 3239 . 2 (𝐴 ∈ V → 𝐴 / 𝑥{⟨𝑦, 𝑧⟩ ∣ 𝜑} = {⟨𝑦, 𝑧⟩ ∣ [𝐴 / 𝑥]𝜑})
12 csbprc 3932 . . 3 𝐴 ∈ V → 𝐴 / 𝑥{⟨𝑦, 𝑧⟩ ∣ 𝜑} = ∅)
13 sbcex 3412 . . . . . . 7 ([𝐴 / 𝑥]𝜑𝐴 ∈ V)
1413con3i 149 . . . . . 6 𝐴 ∈ V → ¬ [𝐴 / 𝑥]𝜑)
1514nexdv 1851 . . . . 5 𝐴 ∈ V → ¬ ∃𝑧[𝐴 / 𝑥]𝜑)
1615nexdv 1851 . . . 4 𝐴 ∈ V → ¬ ∃𝑦𝑧[𝐴 / 𝑥]𝜑)
17 opabn0 4931 . . . . 5 ({⟨𝑦, 𝑧⟩ ∣ [𝐴 / 𝑥]𝜑} ≠ ∅ ↔ ∃𝑦𝑧[𝐴 / 𝑥]𝜑)
1817necon1bbii 2831 . . . 4 (¬ ∃𝑦𝑧[𝐴 / 𝑥]𝜑 ↔ {⟨𝑦, 𝑧⟩ ∣ [𝐴 / 𝑥]𝜑} = ∅)
1916, 18sylib 207 . . 3 𝐴 ∈ V → {⟨𝑦, 𝑧⟩ ∣ [𝐴 / 𝑥]𝜑} = ∅)
2012, 19eqtr4d 2647 . 2 𝐴 ∈ V → 𝐴 / 𝑥{⟨𝑦, 𝑧⟩ ∣ 𝜑} = {⟨𝑦, 𝑧⟩ ∣ [𝐴 / 𝑥]𝜑})
2111, 20pm2.61i 175 1 𝐴 / 𝑥{⟨𝑦, 𝑧⟩ ∣ 𝜑} = {⟨𝑦, 𝑧⟩ ∣ [𝐴 / 𝑥]𝜑}
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   = wceq 1475  ∃wex 1695  [wsb 1867   ∈ wcel 1977  Vcvv 3173  [wsbc 3402  ⦋csb 3499  ∅c0 3874  {copab 4642 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-opab 4644 This theorem is referenced by:  csbmpt12  4934  csbcnv  5228
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