Theorem scottabes 44266

Description: Value of the Scott operation at a class abstraction. Variant of scottab 44265 using explicit substitution. (Contributed by Rohan Ridenour, 14-Aug-2023.)

Assertion

Ref	Expression
scottabes	⊢ Scott {𝑥 ∣ 𝜑} = {𝑥 ∣ (𝜑 ∧ ∀𝑦([𝑦 / 𝑥]𝜑 → (rank‘𝑥) ⊆ (rank‘𝑦)))}

Distinct variable groups:   𝜑,𝑦   𝑥,𝑦

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem scottabes

Step	Hyp	Ref	Expression
1		nfs1v 2155	. 2 ⊢ Ⅎ𝑥[𝑦 / 𝑥]𝜑
2		sbequ12 2250	. 2 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑥]𝜑))
3	1, 2	scottabf 44264	1 ⊢ Scott {𝑥 ∣ 𝜑} = {𝑥 ∣ (𝜑 ∧ ∀𝑦([𝑦 / 𝑥]𝜑 → (rank‘𝑥) ⊆ (rank‘𝑦)))}

Syntax hints:   → wi 4   ∧ wa 395  ∀wal 1537   = wceq 1539  [wsb 2063  {cab 2713   ⊆ wss 3950  ‘cfv 6560  rankcrnk 9804  Scott cscott 44259

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ral 3061  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-iota 6513  df-fv 6568  df-scott 44260

This theorem is referenced by: (None)