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Theorem elscottab 40872
Description: An element of the output of the Scott operation applied to a class abstraction satisfies the class abstraction's predicate. (Contributed by Rohan Ridenour, 14-Aug-2023.)
Hypothesis
Ref Expression
elscottab.1 (𝑥 = 𝑦 → (𝜑𝜓))
Assertion
Ref Expression
elscottab (𝑦 ∈ Scott {𝑥𝜑} → 𝜓)
Distinct variable groups:   𝜓,𝑥   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑦)

Proof of Theorem elscottab
StepHypRef Expression
1 scottss 40871 . . 3 Scott {𝑥𝜑} ⊆ {𝑥𝜑}
21sseli 3949 . 2 (𝑦 ∈ Scott {𝑥𝜑} → 𝑦 ∈ {𝑥𝜑})
3 vex 3483 . . 3 𝑦 ∈ V
4 elscottab.1 . . 3 (𝑥 = 𝑦 → (𝜑𝜓))
53, 4elab 3653 . 2 (𝑦 ∈ {𝑥𝜑} ↔ 𝜓)
62, 5sylib 221 1 (𝑦 ∈ Scott {𝑥𝜑} → 𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wcel 2115  {cab 2802  Scott cscott 40863
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 2179  ax-ext 2796
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-rab 3142  df-v 3482  df-in 3926  df-ss 3936  df-scott 40864
This theorem is referenced by:  cpcolld  40886  grucollcld  40888
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