Theorem elscottab 40654

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 40653	. . 3 ⊢ Scott {𝑥 ∣ 𝜑} ⊆ {𝑥 ∣ 𝜑}
2	1	sseli 3956	. 2 ⊢ (𝑦 ∈ Scott {𝑥 ∣ 𝜑} → 𝑦 ∈ {𝑥 ∣ 𝜑})
3		vex 3494	. . 3 ⊢ 𝑦 ∈ V
4		elscottab.1	. . 3 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
5	3, 4	elab 3663	. 2 ⊢ (𝑦 ∈ {𝑥 ∣ 𝜑} ↔ 𝜓)
6	2, 5	sylib 220	1 ⊢ (𝑦 ∈ Scott {𝑥 ∣ 𝜑} → 𝜓)

Syntax hints:   → wi 4   ↔ wb 208   ∈ wcel 2113  {cab 2798  Scott cscott 40645

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2792

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-clab 2799  df-cleq 2813  df-clel 2892  df-nfc 2962  df-rab 3146  df-v 3493  df-in 3936  df-ss 3945  df-scott 40646

This theorem is referenced by:  cpcolld  40668  grucollcld  40670