Theorem scottabes

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

	Ref	Expression
Assertion	scottabes	$$⊢Scott\left\{x|\phi \right\}=\left\{x|\left(\phi \wedge \forall y\left(\left[y/x\right]\phi \to \mathrm{rank}\left(x\right)\subseteq \mathrm{rank}\left(y\right)\right)\right)\right\}$$

Proof

Step	Hyp	Ref	Expression
1		nfs1v	$$⊢Ⅎx\left[y/x\right]\phi$$
2		sbequ12	$$⊢x=y\to \left(\phi \leftrightarrow \left[y/x\right]\phi \right)$$
3	1 2	scottabf	$$⊢Scott\left\{x|\phi \right\}=\left\{x|\left(\phi \wedge \forall y\left(\left[y/x\right]\phi \to \mathrm{rank}\left(x\right)\subseteq \mathrm{rank}\left(y\right)\right)\right)\right\}$$