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Theorem scottabes 40735
 Description: Value of the Scott operation at a class abstraction. Variant of scottab 40734 using explicit substitution. (Contributed by Rohan Ridenour, 14-Aug-2023.)
Assertion
Ref Expression
scottabes Scott {𝑥𝜑} = {𝑥 ∣ (𝜑 ∧ ∀𝑦([𝑦 / 𝑥]𝜑 → (rank‘𝑥) ⊆ (rank‘𝑦)))}
Distinct variable groups:   𝜑,𝑦   𝑥,𝑦
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem scottabes
StepHypRef Expression
1 nfs1v 2161 . 2 𝑥[𝑦 / 𝑥]𝜑
2 sbequ12 2254 . 2 (𝑥 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑥]𝜑))
31, 2scottabf 40733 1 Scott {𝑥𝜑} = {𝑥 ∣ (𝜑 ∧ ∀𝑦([𝑦 / 𝑥]𝜑 → (rank‘𝑥) ⊆ (rank‘𝑦)))}
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399  ∀wal 1536   = wceq 1538  [wsb 2070  {cab 2799   ⊆ wss 3910  ‘cfv 6328  rankcrnk 9168  Scott cscott 40728 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ral 3131  df-rab 3135  df-v 3473  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4267  df-if 4441  df-sn 4541  df-pr 4543  df-op 4547  df-uni 4812  df-br 5040  df-iota 6287  df-fv 6336  df-scott 40729 This theorem is referenced by: (None)
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