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Metamath Proof Explorer

Theorem rabeq

Description: Equality theorem for restricted class abstractions. (Contributed by NM, 15-Oct-2003) Avoid ax-10 , ax-11 , ax-12 . (Revised by GG, 20-Aug-2023)

		Ref	Expression
	Assertion	rabeq	$⊢ A = B \to \{x \in A \| φ\} = \{x \in B \| φ\}$

Proof

Step	Hyp	Ref	Expression
1		eleq2	$⊢ A = B \to (x \in A \leftrightarrow x \in B)$
2	1	anbi1d	$⊢ A = B \to ((x \in A \land φ) \leftrightarrow (x \in B \land φ))$
3	2	rabbidva2	$⊢ A = B \to \{x \in A \| φ\} = \{x \in B \| φ\}$