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Metamath Proof Explorer

Theorem elabd3

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

	Ref	Expression
Hypotheses	elabd3.ex	$⊢ φ \to A \in V$
	elabd3.is	$⊢ (φ \land x = A) \to (ψ \leftrightarrow χ)$
Assertion	elabd3	$⊢ φ \to (A \in \{x \| ψ\} \leftrightarrow χ)$

Step	Hyp	Ref	Expression
1		elabd3.ex	$⊢ φ \to A \in V$
2		elabd3.is	$⊢ (φ \land x = A) \to (ψ \leftrightarrow χ)$
3		eqidd	$⊢ φ \to \{x \| ψ\} = \{x \| ψ\}$
4	1 3 2	elabd2	$⊢ φ \to (A \in \{x \| ψ\} \leftrightarrow χ)$