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Theorem elscottab 41563
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 41562 . . 3 Scott {𝑥𝜑} ⊆ {𝑥𝜑}
21sseli 3910 . 2 (𝑦 ∈ Scott {𝑥𝜑} → 𝑦 ∈ {𝑥𝜑})
3 vex 3424 . . 3 𝑦 ∈ V
4 elscottab.1 . . 3 (𝑥 = 𝑦 → (𝜑𝜓))
53, 4elab 3599 . 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 2111  {cab 2715  Scott cscott 41554
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2113  ax-9 2121  ax-ext 2709
This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1546  df-ex 1788  df-sb 2072  df-clab 2716  df-cleq 2730  df-clel 2817  df-rab 3071  df-v 3422  df-in 3887  df-ss 3897  df-scott 41555
This theorem is referenced by:  cpcolld  41577  grucollcld  41579
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