Metamath Proof Explorer

Theorem rexabso

Description: Simplification of restricted quantification in a transitive class. When ph is quantifier-free, this shows that the formula E. x e. y ph is absolute for transitive models, which is a particular case of Lemma I.16.2 of Kunen2 p. 95. (Contributed by Eric Schmidt, 19-Oct-2025)

		Ref	Expression
	Assertion	rexabso	$⊢ (Tr M \land A \in M) \to (\exists x \in A φ \leftrightarrow \exists x \in M (x \in A \land φ))$

Proof

Step	Hyp	Ref	Expression
1		trss	$⊢ Tr M \to (A \in M \to A \subseteq M)$
2	1	imp	$⊢ (Tr M \land A \in M) \to A \subseteq M$
3		rexss	$⊢ A \subseteq M \to (\exists x \in A φ \leftrightarrow \exists x \in M (x \in A \land φ))$
4	2 3	syl	$⊢ (Tr M \land A \in M) \to (\exists x \in A φ \leftrightarrow \exists x \in M (x \in A \land φ))$