Theorem scottabes 44399

Description: Value of the Scott operation at a class abstraction. Variant of scottab 44398 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 2161	. 2 ⊢ Ⅎ𝑥[𝑦 / 𝑥]𝜑
2		sbequ12 2256	. 2 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑥]𝜑))
3	1, 2	scottabf 44397	1 ⊢ Scott {𝑥 ∣ 𝜑} = {𝑥 ∣ (𝜑 ∧ ∀𝑦([𝑦 / 𝑥]𝜑 → (rank‘𝑥) ⊆ (rank‘𝑦)))}

Syntax hints:   → wi 4   ∧ wa 395  ∀wal 1539   = wceq 1541  [wsb 2067  {cab 2711   ⊆ wss 3898  ‘cfv 6489  rankcrnk 9667  Scott cscott 44392

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ral 3049  df-rab 3397  df-v 3439  df-dif 3901  df-un 3903  df-ss 3915  df-nul 4283  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4861  df-br 5096  df-iota 6445  df-fv 6497  df-scott 44393

This theorem is referenced by: (None)