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Metamath Proof Explorer

Theorem rabbidva

Description: Equivalent wff's yield equal restricted class abstractions (deduction form). (Contributed by NM, 28-Nov-2003) (Proof shortened by SN, 3-Dec-2023)

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

Proof

Step	Hyp	Ref	Expression
1		rabbidva.1	$⊢ (φ \land x \in A) \to (ψ \leftrightarrow χ)$
2	1	pm5.32da	$⊢ φ \to ((x \in A \land ψ) \leftrightarrow (x \in A \land χ))$
3	2	rabbidva2	$⊢ φ \to \{x \in A \| ψ\} = \{x \in A \| χ\}$