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Metamath Proof Explorer

Theorem rabbiia

Description: Equivalent formulas yield equal restricted class abstractions (inference form). (Contributed by NM, 22-May-1999) (Proof shortened by Wolf Lammen, 12-Jan-2025)

		Ref	Expression
	Hypothesis	rabbiia.1	$⊢ x \in A \to (φ \leftrightarrow ψ)$
	Assertion	rabbiia	$⊢ \{x \in A \| φ\} = \{x \in A \| ψ\}$

Proof

Step	Hyp	Ref	Expression
1		rabbiia.1	$⊢ x \in A \to (φ \leftrightarrow ψ)$
2	1	pm5.32i	$⊢ (x \in A \land φ) \leftrightarrow (x \in A \land ψ)$
3	2	rabbia2	$⊢ \{x \in A \| φ\} = \{x \in A \| ψ\}$