Theorem elscottab

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)

	Ref	Expression
Hypothesis	elscottab.1	$$⊢x=y\to \left(\phi \leftrightarrow \psi \right)$$
Assertion	elscottab	$$⊢y\in Scott\left\{x|\phi \right\}\to \psi$$

Proof

Step	Hyp	Ref	Expression
1		elscottab.1	$$⊢x=y\to \left(\phi \leftrightarrow \psi \right)$$
2		scottss	$$⊢Scott\left\{x|\phi \right\}\subseteq \left\{x|\phi \right\}$$
3	2	sseli	$$⊢y\in Scott\left\{x|\phi \right\}\to y\in \left\{x|\phi \right\}$$
4		vex	$$⊢y\in \mathrm{V}$$
5	4 1	elab	$$⊢y\in \left\{x|\phi \right\}\leftrightarrow \psi$$
6	3 5	sylib	$$⊢y\in Scott\left\{x|\phi \right\}\to \psi$$