Theorem scottab

Description: Value of the Scott operation at a class abstraction. (Contributed by Rohan Ridenour, 14-Aug-2023)

	Ref	Expression
Hypothesis	scottab.1	$$⊢x=y\to \left(\phi \leftrightarrow \psi \right)$$
Assertion	scottab	$$⊢Scott\left\{x|\phi \right\}=\left\{x|\left(\phi \wedge \forall y\left(\psi \to \mathrm{rank}\left(x\right)\subseteq \mathrm{rank}\left(y\right)\right)\right)\right\}$$

Proof

Step	Hyp	Ref	Expression
1		scottab.1	$$⊢x=y\to \left(\phi \leftrightarrow \psi \right)$$
2		nfv	$$⊢Ⅎx\psi$$
3	2 1	scottabf	$$⊢Scott\left\{x|\phi \right\}=\left\{x|\left(\phi \wedge \forall y\left(\psi \to \mathrm{rank}\left(x\right)\subseteq \mathrm{rank}\left(y\right)\right)\right)\right\}$$