Theorem elscottab 43594

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 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
elscottab	⊢ (𝑦 ∈ Scott {𝑥 ∣ 𝜑} → 𝜓)

Distinct variable groups:   𝜓,𝑥   𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑦)

Proof of Theorem elscottab

Step	Hyp	Ref	Expression
1		scottss 43593	. . 3 ⊢ Scott {𝑥 ∣ 𝜑} ⊆ {𝑥 ∣ 𝜑}
2	1	sseli 3974	. 2 ⊢ (𝑦 ∈ Scott {𝑥 ∣ 𝜑} → 𝑦 ∈ {𝑥 ∣ 𝜑})
3		vex 3473	. . 3 ⊢ 𝑦 ∈ V
4		elscottab.1	. . 3 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
5	3, 4	elab 3665	. 2 ⊢ (𝑦 ∈ {𝑥 ∣ 𝜑} ↔ 𝜓)
6	2, 5	sylib 217	1 ⊢ (𝑦 ∈ Scott {𝑥 ∣ 𝜑} → 𝜓)

Syntax hints:   → wi 4   ↔ wb 205   ∈ wcel 2099  {cab 2704  Scott cscott 43585

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-ext 2698

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1537  df-ex 1775  df-sb 2061  df-clab 2705  df-cleq 2719  df-clel 2805  df-rab 3428  df-v 3471  df-in 3951  df-ss 3961  df-scott 43586

This theorem is referenced by:  cpcolld  43608  grucollcld  43610