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Metamath Proof Explorer

Theorem r19.3rzv

Description: Restricted quantification of wff not containing quantified variable. (Contributed by NM, 10-Mar-1997) Avoid ax-12 . (Revised by TM, 16-Feb-2026)

		Ref	Expression
	Assertion	r19.3rzv	$⊢ A \neq \emptyset \to (φ \leftrightarrow \forall x \in A φ)$

Proof

Step	Hyp	Ref	Expression
1		ax-1	$⊢ φ \to (x \in A \to φ)$
2	1	ralrimiv	$⊢ φ \to \forall x \in A φ$
3		rspn0	$⊢ A \neq \emptyset \to (\forall x \in A φ \to φ)$
4	2 3	impbid2	$⊢ A \neq \emptyset \to (φ \leftrightarrow \forall x \in A φ)$