Theorem elabd3

Description: Membership in a class abstraction, using implicit substitution. Deduction version of elab . (Contributed by Gino Giotto, 12-Oct-2024)

	Ref	Expression
Hypotheses	elabd3.ex	$$⊢\phi \to A\in V$$
	elabd3.is	$$⊢\left(\phi \wedge x=A\right)\to \left(\psi \leftrightarrow \chi \right)$$
Assertion	elabd3	$$⊢\phi \to \left(A\in \left\{x|\psi \right\}\leftrightarrow \chi \right)$$

Proof

Step	Hyp	Ref	Expression
1		elabd3.ex	$$⊢\phi \to A\in V$$
2		elabd3.is	$$⊢\left(\phi \wedge x=A\right)\to \left(\psi \leftrightarrow \chi \right)$$
3		eqidd	$$⊢\phi \to \left\{x|\psi \right\}=\left\{x|\psi \right\}$$
4	1 3 2	elabd2	$$⊢\phi \to \left(A\in \left\{x|\psi \right\}\leftrightarrow \chi \right)$$