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Metamath Proof Explorer

Theorem rabidd

Description: An "identity" law of concretion for restricted abstraction. Special case of Definition 2.1 of Quine p. 16. (Contributed by Glauco Siliprandi, 24-Jan-2025)

	Ref	Expression
Hypotheses	rabidd.1	$⊢ φ \to x \in A$
	rabidd.2	$⊢ φ \to χ$
Assertion	rabidd	$⊢ φ \to x \in \{x \in A \| χ\}$

Proof

Step	Hyp	Ref	Expression
1		rabidd.1	$⊢ φ \to x \in A$
2		rabidd.2	$⊢ φ \to χ$
3		rabid	$⊢ x \in \{x \in A \| χ\} \leftrightarrow (x \in A \land χ)$
4	1 2 3	sylanbrc	$⊢ φ \to x \in \{x \in A \| χ\}$