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Metamath Proof Explorer

Theorem rabeqc

Description: A restricted class abstraction equals the restricting class if its condition follows from the membership of the free setvar variable in the restricting class. (Contributed by AV, 20-Apr-2022) (Proof shortened by SN, 15-Jan-2025)

		Ref	Expression
	Hypothesis	rabeqc.1	$⊢ x \in A \to φ$
	Assertion	rabeqc	$⊢ \{x \in A \| φ\} = A$

Proof

Step	Hyp	Ref	Expression
1		rabeqc.1	$⊢ x \in A \to φ$
2	1	adantl	$⊢ (⊤ \land x \in A) \to φ$
3	2	rabeqcda	$⊢ ⊤ \to \{x \in A \| φ\} = A$
4	3	mptru	$⊢ \{x \in A \| φ\} = A$