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Theorem elscottab 42616
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 42615 . . 3 Scott {𝑥𝜑} ⊆ {𝑥𝜑}
21sseli 3944 . 2 (𝑦 ∈ Scott {𝑥𝜑} → 𝑦 ∈ {𝑥𝜑})
3 vex 3451 . . 3 𝑦 ∈ V
4 elscottab.1 . . 3 (𝑥 = 𝑦 → (𝜑𝜓))
53, 4elab 3634 . 2 (𝑦 ∈ {𝑥𝜑} ↔ 𝜓)
62, 5sylib 217 1 (𝑦 ∈ Scott {𝑥𝜑} → 𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wcel 2107  {cab 2710  Scott cscott 42607
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-rab 3407  df-v 3449  df-in 3921  df-ss 3931  df-scott 42608
This theorem is referenced by:  cpcolld  42630  grucollcld  42632
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