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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 ≠ ∅ → ( 𝜑 ↔ ∀ 𝑥 ∈ 𝐴 𝜑 ) )

Proof

Step	Hyp	Ref	Expression
1		ax-1	⊢ ( 𝜑 → ( 𝑥 ∈ 𝐴 → 𝜑 ) )
2	1	ralrimiv	⊢ ( 𝜑 → ∀ 𝑥 ∈ 𝐴 𝜑 )
3		rspn0	⊢ ( 𝐴 ≠ ∅ → ( ∀ 𝑥 ∈ 𝐴 𝜑 → 𝜑 ) )
4	2 3	impbid2	⊢ ( 𝐴 ≠ ∅ → ( 𝜑 ↔ ∀ 𝑥 ∈ 𝐴 𝜑 ) )