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Theorem csbopab 5561
Description: Move substitution into a class abstraction. Version of csbopabgALT 5562 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 3895 . . . 4 (𝑤 = 𝐴𝑤 / 𝑥{⟨𝑦, 𝑧⟩ ∣ 𝜑} = 𝐴 / 𝑥{⟨𝑦, 𝑧⟩ ∣ 𝜑})
2 dfsbcq2 3779 . . . . 5 (𝑤 = 𝐴 → ([𝑤 / 𝑥]𝜑[𝐴 / 𝑥]𝜑))
32opabbidv 5219 . . . 4 (𝑤 = 𝐴 → {⟨𝑦, 𝑧⟩ ∣ [𝑤 / 𝑥]𝜑} = {⟨𝑦, 𝑧⟩ ∣ [𝐴 / 𝑥]𝜑})
41, 3eqeq12d 2742 . . 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 3466 . . . 4 𝑤 ∈ V
6 nfs1v 2146 . . . . 5 𝑥[𝑤 / 𝑥]𝜑
76nfopab 5222 . . . 4 𝑥{⟨𝑦, 𝑧⟩ ∣ [𝑤 / 𝑥]𝜑}
8 sbequ12 2239 . . . . 5 (𝑥 = 𝑤 → (𝜑 ↔ [𝑤 / 𝑥]𝜑))
98opabbidv 5219 . . . 4 (𝑥 = 𝑤 → {⟨𝑦, 𝑧⟩ ∣ 𝜑} = {⟨𝑦, 𝑧⟩ ∣ [𝑤 / 𝑥]𝜑})
105, 7, 9csbief 3927 . . 3 𝑤 / 𝑥{⟨𝑦, 𝑧⟩ ∣ 𝜑} = {⟨𝑦, 𝑧⟩ ∣ [𝑤 / 𝑥]𝜑}
114, 10vtoclg 3534 . 2 (𝐴 ∈ V → 𝐴 / 𝑥{⟨𝑦, 𝑧⟩ ∣ 𝜑} = {⟨𝑦, 𝑧⟩ ∣ [𝐴 / 𝑥]𝜑})
12 csbprc 4411 . . 3 𝐴 ∈ V → 𝐴 / 𝑥{⟨𝑦, 𝑧⟩ ∣ 𝜑} = ∅)
13 sbcex 3786 . . . . . . 7 ([𝐴 / 𝑥]𝜑𝐴 ∈ V)
1413con3i 154 . . . . . 6 𝐴 ∈ V → ¬ [𝐴 / 𝑥]𝜑)
1514nexdv 1932 . . . . 5 𝐴 ∈ V → ¬ ∃𝑧[𝐴 / 𝑥]𝜑)
1615nexdv 1932 . . . 4 𝐴 ∈ V → ¬ ∃𝑦𝑧[𝐴 / 𝑥]𝜑)
17 opabn0 5559 . . . . 5 ({⟨𝑦, 𝑧⟩ ∣ [𝐴 / 𝑥]𝜑} ≠ ∅ ↔ ∃𝑦𝑧[𝐴 / 𝑥]𝜑)
1817necon1bbii 2980 . . . 4 (¬ ∃𝑦𝑧[𝐴 / 𝑥]𝜑 ↔ {⟨𝑦, 𝑧⟩ ∣ [𝐴 / 𝑥]𝜑} = ∅)
1916, 18sylib 217 . . 3 𝐴 ∈ V → {⟨𝑦, 𝑧⟩ ∣ [𝐴 / 𝑥]𝜑} = ∅)
2012, 19eqtr4d 2769 . 2 𝐴 ∈ V → 𝐴 / 𝑥{⟨𝑦, 𝑧⟩ ∣ 𝜑} = {⟨𝑦, 𝑧⟩ ∣ [𝐴 / 𝑥]𝜑})
2111, 20pm2.61i 182 1 𝐴 / 𝑥{⟨𝑦, 𝑧⟩ ∣ 𝜑} = {⟨𝑦, 𝑧⟩ ∣ [𝐴 / 𝑥]𝜑}
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1534  wex 1774  [wsb 2060  wcel 2099  Vcvv 3462  [wsbc 3776  csb 3892  c0 4325  {copab 5215
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-sep 5304  ax-nul 5311  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-v 3464  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-ss 3964  df-nul 4326  df-if 4534  df-sn 4634  df-pr 4636  df-op 4640  df-opab 5216
This theorem is referenced by:  csbmpt12  5563  csbcnv  5890
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