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Metamath Proof Explorer

Theorem eqabi

Description: Equality of a class variable and a class abstraction (inference form). (Contributed by NM, 26-May-1993) Avoid ax-11 . (Revised by Wolf Lammen, 6-May-2023)

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

Proof

Step	Hyp	Ref	Expression
1		eqabi.1	$⊢ x \in A \leftrightarrow φ$
2	1	a1i	$⊢ ⊤ \to (x \in A \leftrightarrow φ)$
3	2	eqabdv	$⊢ ⊤ \to A = \{x \| φ\}$
4	3	mptru	$⊢ A = \{x \| φ\}$